import { type PracticeQuestion } from "../../types/lesson";

export const NONLINEAR_EQ_EASY: PracticeQuestion[] = [
  {
    id: "1e003284",
    type: "mcq",
    questionHtml:
      "<strong>x = 49</strong><br><strong>y = √(x) + 9</strong><br>The graphs of the given equations intersect at the point <strong>(x, y)</strong> in the xy-plane. What is the value of <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>16</strong>" },
      { label: "B", text: "<strong>40</strong>" },
      { label: "C", text: "<strong>81</strong>" },
      { label: "D", text: "<strong>130</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the graphs of the given equations intersect at the point <strong>(x, y)</strong> in the xy-plane. It follows that <strong>(x, y)</strong> represents a solution to the system consisting of the given equations. The first equation given is <strong>x = 49</strong>. Substituting <strong>49</strong> for <strong>x</strong> in the second equation given, <strong>y = √(x) + 9</strong>, yields <strong>y = √(49) + 9</strong>, which is equivalent to <strong>y = 7 + 9</strong>, or <strong>y = 16</strong>. It follows that the graphs of the given equations intersect at the point <strong>(49, 16)</strong>. Therefore, the value of <strong>y</strong> is <strong>16</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "3c95093c",
    type: "mcq",
    questionHtml:
      "Which of the following inequalities is equivalent to the inequality above?",
    choices: [
      { label: "A", text: "<strong>x − y > 2</strong>" },
      { label: "B", text: "<strong>2 x − 3 y > 4</strong>" },
      { label: "C", text: "<strong>3 x − 2 y > 4</strong>" },
      { label: "D", text: "<strong>3 y − 2 x > 2</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Both sides of the given inequality can be divided by 3 to yield <strong>2 x − 3 y > 4</strong>.Choices A, C, and D are incorrect because they are not equivalent to (do not have the same solution set as) the given inequality. For example, the ordered pair <strong>0 −1 . 5</strong> is a solution to the given inequality, but it is not a solution to any of the inequalities in choices A, C, or D.",
    hasFigure: false,
  },
  {
    id: "3de7a7d7",
    type: "mcq",
    questionHtml:
      "Which of the following is a solution to the equation <strong>2 x² − 4 = x²</strong> ?",
    choices: [
      { label: "A", text: "1" },
      { label: "B", text: "2" },
      { label: "C", text: "3" },
      { label: "D", text: "4" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Subtracting x2 from both sides of the given equation yields x2 – 4 = 0. Adding 4 to both sides of the equation gives x2 = 4. Taking the square root of both sides of the equation gives x = 2 or x = –2. Therefore, x = 2 is one solution to the original equation.Alternative approach: Subtracting x2 from both sides of the given equation yields x2 – 4 = 0. Factoring this equation gives x2 – 4 = (x + 2)(x – 2) = 0, such that x = 2 or x = –2. Therefore, x = 2 is one solution to the original equation.<br>Choices A, C, and D are incorrect and may be the result of computation errors.",
    hasFigure: false,
  },
  {
    id: "4236c5a3",
    type: "mcq",
    questionHtml:
      "If <strong>(x + 5, ), ² = 4</strong>, which of the following is a possible value of x ?",
    choices: [
      { label: "A", text: "1" },
      { label: "B", text: "<strong>−1</strong>" },
      { label: "C", text: "<strong>−2</strong>" },
      { label: "D", text: "<strong>−3</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. If <strong>(x + 5, ), ² = 4</strong>, then taking the square root of each side of the equation gives <strong>x + 5 = 2</strong> or <strong>x + 5 = −2</strong>. Solving these equations for x gives <strong>x = −3</strong> or <strong>x = −7</strong>. Of these, <strong>−3</strong> is the only solution given as a choice.Choice A is incorrect and may result from solving the equation <strong>x + 5 = 4</strong> and making a sign error. Choice B is incorrect and may result from solving the equation <strong>x + 5 = 4</strong>. Choice C is incorrect and may result from finding a possible value of <strong>x + 5</strong>.",
    hasFigure: false,
  },
  {
    id: "4ca30186",
    type: "mcq",
    questionHtml:
      "For the curve in the system:<br><br>Moving from left to right:<br><br>The curve passes from quadrant 2 to quadrant 1.<br>In quadrant 2, the curve trends up gradually to point (0 comma 3.25).<br>In quadrant 1, the curve trends up sharply.<br><br>The curve passes through the following points:<br><br>(1 comma 3.5)<br>(2 comma 4)<br>(4 comma 7)<br><br>For the line in the system:<br><br>The line slants gradually up from left to right.<br>The line passes through the following points:<br><br>(2 comma 4)<br>(7 comma 7.5)<br><br>The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution <strong>(x, y)</strong> to this system?",
    choices: [
      { label: "A", text: "<strong>(0, 0)</strong>" },
      { label: "B", text: "<strong>(0, 2)</strong>" },
      { label: "C", text: "<strong>(2, 4)</strong>" },
      { label: "D", text: "<strong>(4, 0)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point <strong>(2, 4)</strong>. Thus, the solution to the system is <strong>(2, 4)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/4ca30186_svg1.svg",
  },
  {
    id: "568aaf27",
    type: "mcq",
    questionHtml:
      "If <strong>the ordered pair x, y</strong> is a solution to the system of equations above, which of the following is a possible value of x?",
    choices: [
      { label: "A", text: "0" },
      { label: "B", text: "1" },
      { label: "C", text: "2" },
      { label: "D", text: "3" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Substituting <strong>x²</strong> from the second equation for y in the first equation yields <strong>x + x² = 12</strong>. Subtracting 12 from both sides of this equation and rewriting the equation results in <strong>x² + x − 12 = 0</strong>. Factoring the left-hand side of this equation yields <strong>(x − 3, ) · (x + 4, ) = 0</strong>. Using the zero product property to solve for x, it follows that <strong>x − 3 = 0</strong> and <strong>x + 4 = 0</strong>. Solving each equation for x yields <strong>x = 3</strong> and <strong>x = −4</strong>, respectively. Thus, two possible values of x are 3 and <strong>−4</strong>. Of the choices given, 3 is the only possible value of x.Choices A, B, and C are incorrect. Substituting 0 for x in the first equation gives <strong>0 + y = 12</strong>, or <strong>y = 12</strong>; then, substituting 12 for y and 0 for x in the second equation gives <strong>12 = 0²</strong>, or <strong>12 = 0</strong>, which is false. Similarly, substituting 1 or 2 for x in the first equation yields <strong>y = 11</strong> or <strong>y = 10</strong>, respectively; then, substituting 11 or 10 for y in the second equation yields a false statement.",
    hasFigure: false,
  },
  {
    id: "70f98ab4",
    type: "mcq",
    questionHtml:
      "<strong>q − 29 r = s</strong><br>The given equation relates the positive numbers <strong>q</strong>, <strong>r</strong>, and <strong>s</strong>. Which equation correctly expresses <strong>q</strong> in terms of <strong>r</strong> and <strong>s</strong>?",
    choices: [
      { label: "A", text: "<strong>q = s − 29 r</strong>" },
      { label: "B", text: "<strong>q = s + 29 r</strong>" },
      { label: "C", text: "<strong>q = 29 r s</strong>" },
      { label: "D", text: "<strong>q = − (s) / (29 r)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Adding <strong>29 r</strong> to each side of the given equation yields <strong>q = s + 29 r</strong>. Therefore, the equation <strong>q = s + 29 r</strong> correctly expresses <strong>q</strong> in terms of <strong>r</strong> and <strong>s</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "7cb3a8ee",
    type: "spr",
    questionHtml:
      "<strong>|x − 5| = 10</strong><br>What is one possible solution to the given equation?",
    choices: [],
    correctAnswer: "15, -5",
    explanation:
      "The correct answer is <strong>15</strong> or <strong>−5</strong>. By the definition of absolute value, if <strong>|x − 5| = 10</strong>, then <strong>x − 5 = 10</strong> or <strong>x − 5 = −10</strong>. Adding <strong>5</strong> to both sides of the first equation yields <strong>x = 15</strong>. Adding <strong>5</strong> to both sides of the second equation yields <strong>x = −5</strong>. Thus, the given equation has two possible solutions, <strong>15</strong> and <strong>−5</strong>. Note that 15 and -5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "84e5e36c",
    type: "mcq",
    questionHtml:
      "<strong>y = 76</strong><br><strong>y = x² − 5</strong><br>The graphs of the given equations in the xy-plane intersect at the point <strong>(x, y)</strong>. What is a possible value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>−(76) / (5)</strong>" },
      { label: "B", text: "<strong>−9</strong>" },
      { label: "C", text: "<strong>5</strong>" },
      { label: "D", text: "<strong>76</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since the point <strong>(x, y)</strong> is an intersection point of the graphs of the given equations in the xy-plane, the pair <strong>(x, y)</strong> should satisfy both equations, and thus is a solution of the given system. According to the first equation, <strong>y = 76</strong>. Substituting <strong>76</strong> in place of <strong>y</strong> in the second equation yields <strong>x² − 5 = 76</strong>. Adding <strong>5</strong> to both sides of this equation yields <strong>x² = 81</strong>. Taking the square root of both sides of this equation yields two solutions: <strong>x = 9</strong> and <strong>x = −9</strong>. Of these two solutions, only <strong>−9</strong> is given as a choice.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the value of coordinate <strong>y</strong>, rather than <strong>x</strong>, of the intersection point <strong>(x, y)</strong>.",
    hasFigure: false,
  },
  {
    id: "98f735f2",
    type: "mcq",
    questionHtml:
      "The total revenue from sales of a product can be calculated using the formula <strong>T = P Q</strong>, where T is the total revenue, P is the price of the product, and Q is the quantity of the product sold. Which of the following equations gives the quantity of product sold in terms of P and T ?",
    choices: [
      { label: "A", text: "<strong>Q = P over T</strong>" },
      { label: "B", text: "<strong>Q = T over P</strong>" },
      { label: "C", text: "<strong>Q = P T</strong>" },
      { label: "D", text: "<strong>Q = T − P</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Solving the given equation for Q gives the quantity of the product sold in terms of P and T. Dividing both sides of the given equation by P yields <strong>T over P = Q</strong>, or <strong>Q = T over P</strong>. Therefore, <strong>Q = T over P</strong> gives the quantity of product sold in terms of P and T.Choice A is incorrect and may result from an error when dividing both sides of the given equation by P. Choice C is incorrect and may result from multiplying, rather than dividing, both sides of the given equation by P. Choice D is incorrect and may result from subtracting P from both sides of the equation rather than dividing both sides by P.",
    hasFigure: false,
  },
  {
    id: "a67a439d",
    type: "mcq",
    questionHtml:
      "<strong>x + 7 = 10</strong><br><strong>(x + 7)² = y</strong><br>Which ordered pair <strong>(x, y)</strong> is a solution to the given system of equations?",
    choices: [
      { label: "A", text: "<strong>(3, 100)</strong>" },
      { label: "B", text: "<strong>(3, 3)</strong>" },
      { label: "C", text: "<strong>(3, 10)</strong>" },
      { label: "D", text: "<strong>(3, 70)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The solution to a system of equations is the ordered pair <strong>(x, y)</strong> that satisfies all equations in the system. It's given by the first equation in the system that <strong>x + 7 = 10</strong>. Substituting <strong>10</strong> for <strong>x + 7</strong> into the second equation yields <strong>10² = y</strong>, or <strong>y = 100</strong>. The x-coordinate of the solution to the system of equations can be found by subtracting <strong>7</strong> from both sides of the equation <strong>x + 7 = 10</strong>, which yields <strong>x = 3</strong>. Therefore, the ordered pair <strong>(3, 100)</strong> is a solution to the given system of equations.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "ad03127d",
    type: "mcq",
    questionHtml:
      "<strong>6 r = 7 s + t</strong><br>The given equation relates the variables <strong>r</strong>, <strong>s</strong>, and <strong>t</strong>. Which equation correctly expresses <strong>s</strong> in terms of <strong>r</strong> and <strong>t</strong>?",
    choices: [
      { label: "A", text: "<strong>s = 42 r − t</strong>" },
      { label: "B", text: "<strong>s = 7 (6 r − t)</strong>" },
      { label: "C", text: "<strong>s = six sevenths r − t</strong>" },
      { label: "D", text: "<strong>s = (6 r − t) / (7)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Subtracting <strong>t</strong> from both sides of the given equation yields <strong>6 r − t = 7 s</strong>. Dividing both sides of this equation by <strong>7</strong> yields <strong>(6 r − t) / (7) = s</strong>. Therefore, the equation <strong>s = (6 r − t) / (7)</strong> correctly expresses <strong>s</strong> in terms of <strong>r</strong> and <strong>t</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b76a2815",
    type: "mcq",
    questionHtml:
      "The power P produced by a machine is represented by the equation above, where W is the work performed during an amount of time t. Which of the following correctly expresses W in terms of P and t ?",
    choices: [
      { label: "A", text: "<strong>W = P t</strong>" },
      { label: "B", text: "<strong>W = the fraction P over t</strong>" },
      { label: "C", text: "<strong>W = the fraction t over P</strong>" },
      { label: "D", text: "<strong>W = P + t</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Multiplying both sides of the equation by t yields <strong>P · t = (the fraction w over t, ) · t</strong>, or <strong>P t = W</strong>, which expresses W in terms of P and t. This is equivalent to W = Pt.<br> <br><br>Choices B, C, and D are incorrect. Each of the expressions given in these answer choices gives W in terms of P and t but doesn’t maintain the given relationship between W, P, and t. These expressions may result from performing different operations with t on each side of the equation. In choice B, W has been multiplied by t, and P has been divided by t. In choice C, W has been multiplied by t, and the quotient of P divided by t has been reciprocated. In choice D, W has been multiplied by t, and P has been added to t. However, in order to maintain the relationship between the variables in the given equation, the same operation must be performed with t on each side of the equation.",
    hasFigure: false,
  },
  {
    id: "b8c4a1cd",
    type: "mcq",
    questionHtml:
      "<strong>8 j = k + 15 m</strong><br>The given equation relates the distinct positive numbers <strong>j</strong>, <strong>k</strong>, and <strong>m</strong>. Which equation correctly expresses <strong>j</strong> in terms of <strong>k</strong> and <strong>m</strong>?",
    choices: [
      { label: "A", text: "<strong>j = (k) / (8) + 15 m</strong>" },
      { label: "B", text: "<strong>j = k + (15 m) / (8)</strong>" },
      { label: "C", text: "<strong>j = 8 (k + 15 m)</strong>" },
      { label: "D", text: "<strong>j = (k + 15 m) / (8)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. To express <strong>j</strong> in terms of <strong>k</strong> and <strong>m</strong>, the given equation must be solved for <strong>j</strong>. Dividing each side of the given equation by <strong>8</strong> yields  <strong>j = (k + 15 m) / (8)</strong>.<br>Choice A is incorrect. This is equivalent to <strong>8 j = k + 120 m</strong>.<br>Choice B is incorrect. This is equivalent to <strong>8 j = 8 k + 15 m</strong>.<br>Choice C is incorrect. This is equivalent to <strong>(j) / (8) = k + 15 m</strong>.",
    hasFigure: false,
  },
  {
    id: "c7789423",
    type: "spr",
    questionHtml:
      "<strong>|x − 2| = 9</strong><br>What is one possible solution to the given equation?",
    choices: [],
    correctAnswer: "11, -7",
    explanation:
      "The correct answer is <strong>11</strong> or <strong>−7</strong>. By the definition of absolute value, if <strong>|x − 2| = 9</strong>, then <strong>x − 2 = 9</strong> or <strong>x − 2 = −9</strong>. Adding <strong>2</strong> to both sides of the equation <strong>x − 2 = 9</strong> yields <strong>x = 11</strong>. Adding <strong>2</strong> to both sides of the equation <strong>x − 2 = −9</strong> yields <strong>x = −7</strong>. Thus, the given equation, <strong>|x − 2| = 9</strong>, has two possible solutions, <strong>11</strong> and <strong>−7</strong>. Note that 11 and -7 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "c8bf5313",
    type: "mcq",
    questionHtml:
      "<strong>x = 8</strong><br><strong>y = x² + 8</strong><br>The graphs of the equations in the given system of equations intersect at the point <strong>(x, y)</strong> in the xy-plane. What is the value of <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>8</strong>" },
      { label: "B", text: "<strong>24</strong>" },
      { label: "C", text: "<strong>64</strong>" },
      { label: "D", text: "<strong>72</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since the graphs of the equations in the given system intersect at the point <strong>(x, y)</strong>, the point <strong>(x, y)</strong> represents a solution to the given system of equations. The first equation of the given system of equations states that <strong>x = 8</strong>. Substituting <strong>8</strong> for <strong>x</strong> in the second equation of the given system of equations yields <strong>y = 8² + 8</strong>, or <strong>y = 72</strong>. Therefore, the value of <strong>y</strong> is <strong>72</strong>.<br>Choice A is incorrect. This is the value of <strong>x</strong>, not <strong>y</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "ce940f80",
    type: "mcq",
    questionHtml:
      "<strong>(x²) / (25) = 36</strong><br>What is a solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>6</strong>" },
      { label: "B", text: "<strong>30</strong>" },
      { label: "C", text: "<strong>450</strong>" },
      { label: "D", text: "<strong>900</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Multiplying the left- and right-hand sides of the given equation by <strong>25</strong> yields <strong>x² = 900</strong>. Taking the square root of the left- and right-hand sides of this equation yields <strong>x = 30</strong> or <strong>x = −30</strong>. Of these two solutions, only <strong>30</strong> is given as a choice.<br>Choice A is incorrect. This is a solution to the equation <strong>x² = 36</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "eb268057",
    type: "mcq",
    questionHtml:
      "Which of the following values of x satisfies the given equation?",
    choices: [
      { label: "A", text: "<strong>−8</strong>" },
      { label: "B", text: "<strong>4</strong>" },
      { label: "C", text: "<strong>32</strong>" },
      { label: "D", text: "<strong>128</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Solving for x by taking the square root of both sides of the given equation yields <strong>x = 8</strong> or <strong>x = −8</strong>. Of the choices given, <strong>−8</strong> satisfies the given equation.Choice B is incorrect and may result from a calculation error when solving for x. Choice C is incorrect and may result from dividing 64 by 2 instead of taking the square root. Choice D is incorrect and may result from multiplying 64 by 2 instead of taking the square root.",
    hasFigure: false,
  },
  {
    id: "f11ffa93",
    type: "spr",
    questionHtml: "What value of x satisfies the equation above?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 117. Squaring both sides of the given equation gives <strong>x + 4 = 11²</strong>, or <strong>x + 4 = 121</strong>. Subtracting 4 from both sides of this equation gives <strong>x = 117</strong>.",
    hasFigure: false,
  },
  {
    id: "fcb78856",
    type: "mcq",
    questionHtml:
      "<strong>b = 42 c f</strong><br>The given equation relates the positive numbers <strong>b</strong>, <strong>c</strong>, and <strong>f</strong>. Which equation correctly expresses <strong>c</strong> in terms of <strong>b</strong> and <strong>f</strong>?",
    choices: [
      { label: "A", text: "<strong>c = (b) / (42 f)</strong>" },
      { label: "B", text: "<strong>c = (b − 42) / (f)</strong>" },
      { label: "C", text: "<strong>c = 42 b f</strong>" },
      { label: "D", text: "<strong>c = 42 − b − f</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the equation <strong>b = 42 c f</strong> relates the positive numbers <strong>b</strong>, <strong>c</strong>, and <strong>f</strong>. Dividing each side of the given equation by <strong>42 f</strong> yields <strong>(b) / (42 f) = c</strong>, or <strong>c = (b) / (42 f)</strong>. Thus, the equation <strong>c = (b) / (42 f)</strong> correctly expresses <strong>c</strong> in terms of <strong>b</strong> and <strong>f</strong>.<br>Choice B is incorrect. This equation can be rewritten as <strong>b = c f + 42</strong>.<br>Choice C is incorrect. This equation can be rewritten as <strong>b = (c) / (42 f)</strong>.<br>Choice D is incorrect. This equation can be rewritten as <strong>b = 42 − c − f</strong>.",
    hasFigure: false,
  },
];

export const NONLINEAR_EQ_MEDIUM: PracticeQuestion[] = [
  {
    id: "062f86db",
    type: "mcq",
    questionHtml:
      "<strong>5 x² − 37 x − 24 = 0</strong><br>What is the positive solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>three fifths</strong>" },
      { label: "B", text: "<strong>3</strong>" },
      { label: "C", text: "<strong>8</strong>" },
      { label: "D", text: "<strong>37</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The left-hand side of the given equation can be factored as <strong>(5 x + 3) (x − 8)</strong>. Therefore, the given equation, <strong>5 x² − 37 x − 24 = 0</strong>, can be written as <strong>(5 x + 3) (x − 8) = 0</strong>. Applying the zero product property to this equation yields <strong>5 x + 3 = 0</strong> and <strong>x − 8 = 0</strong>. Subtracting <strong>3</strong> from both sides of the equation <strong>5 x + 3 = 0</strong> yields <strong>5 x = −3</strong>. Dividing both sides of this equation by <strong>5</strong> yields <strong>x = −three fifths</strong>. Adding <strong>8</strong> to both sides of the equation <strong>x − 8 = 0</strong> yields <strong>x = 8</strong>. Therefore, the two solutions to the given equation, <strong>5 x² − 37 x − 24 = 0</strong>, are <strong>−three fifths</strong> and <strong>8</strong>. It follows that <strong>8</strong> is the positive solution to the given equation.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "0980fcdd",
    type: "mcq",
    questionHtml:
      "A solution to the given system of equations is <strong>the ordered pair x, y</strong>. Which of the following is a possible value of xy ?",
    choices: [
      { label: "A", text: "0" },
      { label: "B", text: "6" },
      { label: "C", text: "12" },
      { label: "D", text: "36" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Solutions to the given system of equations are ordered pairs <strong>x, y</strong> that satisfy both equations in the system. Adding the left-hand and right-hand sides of the equations in the system yields <strong>x² + y = 6 x + −6 x + y + 36</strong>, or <strong>x² + y = y + 36</strong>. Subtracting y from both sides of this equation yields <strong>x² = 36</strong>. Taking the square root of both sides of this equation yields <strong>x = 6</strong> and <strong>x = −6</strong>. Therefore, there are two solutions to this system of equations, one with an x-coordinate of 6 and the other with an x-coordinate of <strong>−6</strong>. Substituting 6 for x in the second equation yields <strong>y = −6 · 6 + 36</strong>, or <strong>y = 0</strong>; therefore, one solution is <strong>the ordered pair 6, 0</strong>. Similarly, substituting <strong>−6</strong> for x in the second equation yields <strong>y = −6 · −6 + 36</strong>, or <strong>y = 72</strong>; therefore, the other solution is <strong>the ordered pair − 6, 72</strong>. It follows then that if <strong>the ordered pair x, y</strong> is a solution to the system, then possible values of <strong>xy</strong> are <strong>6 · 0 = 0</strong> and <strong>−6 · 72 = −432</strong>. Only 0 is among the given choices.Choice B is incorrect. This is the x-coordinate of one of the solutions, <strong>the ordered pair 6, 0</strong>. Choice C is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This is the square of the x-coordinate of one of the solutions, <strong>the ordered pair 6, 0</strong>.",
    hasFigure: false,
  },
  {
    id: "11ccf3e1",
    type: "mcq",
    questionHtml:
      "<strong>14 j + 5 k = m</strong><br>The given equation relates the numbers <strong>j</strong>, <strong>k</strong>, and <strong>m</strong>. Which equation correctly expresses <strong>k</strong> in terms of <strong>j</strong> and <strong>m</strong>?",
    choices: [
      { label: "A", text: "<strong>k = (m − 14 j) / (5)</strong>" },
      { label: "B", text: "<strong>k = one fifth m − 14 j</strong>" },
      { label: "C", text: "<strong>k = (14 j − m) / (5)</strong>" },
      { label: "D", text: "<strong>k = 5 m − 14 j</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Subtracting <strong>14 j</strong> from each side of the given equation results in <strong>5 k = m − 14 j</strong>. Dividing each side of this equation by <strong>5</strong> results in <strong>k = (m − 14 j) / (5)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "13e57f0a",
    type: "mcq",
    questionHtml:
      "<strong>− 4 x² − 7 x = −36</strong><br>What is the positive solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>seven fourths</strong>" },
      { label: "B", text: "<strong>nine fourths</strong>" },
      { label: "C", text: "<strong>4</strong>" },
      { label: "D", text: "<strong>7</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Multiplying each side of the given equation by <strong>−16</strong> yields <strong>64 x² + 112 x = 576</strong>. To complete the square, adding <strong>49</strong> to each side of this equation yields <strong>64 x² + 112 x + 49 = 576 + 49</strong>, or <strong>(8 x + 7)² = 625</strong>. Taking the square root of each side of this equation yields two equations: <strong>8 x + 7 = 25</strong> and <strong>8 x + 7 = −25</strong>. Subtracting <strong>7</strong> from each side of the equation <strong>8 x + 7 = 25</strong> yields <strong>8 x = 18</strong>. Dividing each side of this equation by <strong>8</strong> yields <strong>x = (18) / (8)</strong>, or <strong>x = nine fourths</strong>. Therefore, <strong>nine fourths</strong> is a solution to the given equation. Subtracting <strong>7</strong> from each side of the equation <strong>8 x + 7 = −25</strong> yields <strong>8 x = −32</strong>. Dividing each side of this equation by <strong>8</strong> yields <strong>x = −4</strong>. Therefore, the given equation has two solutions, <strong>nine fourths</strong> and <strong>−4</strong>. Since <strong>nine fourths</strong> is positive, it follows that <strong>nine fourths</strong> is the positive solution to the given equation.<br>Alternate approach: Adding <strong>4 x²</strong> and <strong>7 x</strong> to each side of the given equation yields <strong>0 = 4 x² + 7 x − 36</strong>. The right-hand side of this equation can be rewritten as <strong>4 x² + 16 x − 9 x − 36</strong>. Factoring out the common factor of <strong>4 x</strong> from the first two terms of this expression and the common factor of <strong>−9</strong> from the second two terms yields <strong>4 x (x + 4) − 9 (x + 4)</strong>. Factoring out the common factor of <strong>(x + 4)</strong> from these two terms yields the expression <strong>(4 x − 9) (x + 4)</strong>. Since this expression is equal to <strong>0</strong>, it follows that either <strong>4 x − 9 = 0</strong> or <strong>x + 4 = 0</strong>. Adding <strong>9</strong> to each side of the equation <strong>4 x − 9 = 0</strong> yields <strong>4 x = 9</strong>. Dividing each side of this equation by <strong>4</strong> yields <strong>x = nine fourths</strong>. Therefore, <strong>nine fourths</strong> is a positive solution to the given equation. Subtracting <strong>4</strong> from each side of the equation <strong>x + 4 = 0</strong> yields <strong>x = −4</strong>. Therefore, the given equation has two solutions, <strong>nine fourths</strong> and <strong>−4</strong>. Since <strong>nine fourths</strong> is positive, it follows that <strong>nine fourths</strong> is the positive solution to the given equation.<br>Choice A is incorrect. Substituting <strong>seven fourths</strong> for <strong>x</strong> in the given equation yields <strong>−(49) / (2) = −36</strong>, which is false.<br>Choice C is incorrect. Substituting <strong>4</strong> for <strong>x</strong> in the given equation yields <strong>−92 = −36</strong>, which is false.<br>Choice D is incorrect. Substituting <strong>7</strong> for <strong>x</strong> in the given equation yields <strong>−245 = −36</strong>, which is false.",
    hasFigure: false,
  },
  {
    id: "2683b5db",
    type: "mcq",
    questionHtml:
      "In a city, the property tax T, in dollars, is calculated using the formula above, where P is the value of the property, in dollars. Which of the following expresses the value of the property in terms of the property tax?",
    choices: [
      { label: "A", text: "<strong>P = 100 T − 400</strong>" },
      { label: "B", text: "<strong>P = 100 T + 400</strong>" },
      { label: "C", text: "<strong>P = 100 T − 40, 000</strong>" },
      { label: "D", text: "<strong>P = 100 T + 40, 000</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. To express the value of the property in terms of the property tax, the given equation must be solved for P. Multiplying both sides of the equation by 100 gives <strong>100 T = P − 40, 000</strong>. Adding 40,000 to both sides of the equation gives <strong>100 T + 40, 000 = P</strong>. Therefore, <strong>P = 100 T + 40, 000</strong>.Choice A is incorrect and may result from multiplying 40,000 by 0.01, then subtracting 400 from, instead of adding 400 to, the left-hand side of the equation. Choice B is incorrect and may result from multiplying 40,000 by 0.01. Choice C is incorrect and may result from subtracting instead of adding 40,000 from the left-hand side of the equation.",
    hasFigure: false,
  },
  {
    id: "29ed5d39",
    type: "mcq",
    questionHtml:
      "<strong>p = 20 + (16) / (n)</strong><br>The given equation relates the numbers <strong>p</strong> and <strong>n</strong>, where <strong>n</strong> is not equal to <strong>0</strong> and <strong>p > 20</strong>. Which equation correctly expresses <strong>n</strong> in terms of <strong>p</strong>?",
    choices: [
      { label: "A", text: "<strong>n = (p − 20) / (16)</strong>" },
      { label: "B", text: "<strong>n = (p) / (16) + 20</strong>" },
      { label: "C", text: "<strong>n = (p) / (16) − 20</strong>" },
      { label: "D", text: "<strong>n = (16) / (p − 20)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. To express <strong>n</strong> in terms of <strong>p</strong>, the given equation must be solved for <strong>n</strong>. Subtracting <strong>9</strong> from both sides of the given equation yields <strong>p − 9 = (14) / (n)</strong>. Since <strong>n</strong> is not equal to <strong>0</strong>, multiplying both sides of this equation by <strong>n</strong> yields <strong>(p − 9) (n) = 14</strong>. It's given that <strong>p > 9</strong>, which means <strong>p − 9</strong> is not equal to <strong>0</strong>. Therefore, dividing both sides of <strong>(p − 9) (n) = 14</strong> by <strong>(p − 9)</strong> yields <strong>((p − 9) (n)) / (p − 9) = (14) / (p − 9)</strong>, or <strong>n = (14) / (p − 9)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "2d2ab76b",
    type: "mcq",
    questionHtml:
      "When the equations above are graphed in the xy-plane, what are the coordinates (x, y) of the points of intersection of the two graphs?",
    choices: [
      { label: "A", text: "<strong>2, 3</strong> and <strong>−2, 3</strong>" },
      { label: "B", text: "<strong>2, 4</strong> and <strong>−2, 4</strong>" },
      { label: "C", text: "<strong>3, 8</strong> and <strong>−3, 8</strong>" },
      {
        label: "D",
        text: "<strong>the √ 2, 3</strong> and <strong>the − √ 2, 3</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The two equations form a system of equations, and the solutions to the system correspond to the points of intersection of the graphs. The solutions to the system can be found by substitution. Since the second equation gives y = 3, substituting 3 for y in the first equation gives 3 = x2 – 1. Adding 1 to both sides of the equation gives 4 = x2. Solving by taking the square root of both sides of the equation gives x = ±2. Since y = 3 for all values of x for the second equation, the solutions are (2, 3) and (–2, 3). Therefore, the points of intersection of the two graphs are (2, 3) and (–2, 3).Choices B, C, and D are incorrect and may be the result of calculation errors.",
    hasFigure: false,
  },
  {
    id: "2f958af9",
    type: "mcq",
    questionHtml:
      "<strong>v² = the fraction with numerator L T, and denominator m</strong><br> The formula above expresses the square of the speed v of a wave moving along a string in terms of tension T, mass m, and length L of the string. What is T in terms of m, v, and L ?",
    choices: [
      {
        label: "A",
        text: "<strong>T = the fraction with numerator m v², and denominator L</strong>",
      },
      {
        label: "B",
        text: "<strong>T = the fraction with numerator m, and denominator v² L, end fraction</strong>",
      },
      {
        label: "C",
        text: "<strong>T = the fraction with numerator m L, and denominator v², end fraction</strong>",
      },
      {
        label: "D",
        text: "<strong>T = the fraction with numerator L, and denominator m v², end fraction</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. To write the formula as T in terms of m, v, and L means to isolate T on one side of the equation. First, multiply both sides of the equation by m, which gives <strong>m v² = the fraction with numerator m L T, and denominator m</strong>, which simplifies to mv2 = LT. Next, divide both sides of the equation by L, which gives <strong>the fraction with numerator m v², and denominator L = the fraction with numerator L T, and denominator L</strong>, which simplifies to <strong>T = the fraction with numerator m v², and denominator L</strong>.Choices B, C, and D are incorrect and may be the result of incorrectly applying operations to each side of the equation.",
    hasFigure: false,
  },
  {
    id: "364a2d25",
    type: "spr",
    questionHtml:
      "If one solution to the system of equations above is <strong>the ordered pair x, y</strong>, what is one possible value of x ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is either 8 or 9. The first equation can be rewritten as <strong>y = 17 − x</strong>. Substituting <strong>17 − x</strong> for y in the second equation gives <strong>x · (17 − x, ) = 72</strong>. By applying the distributive property, this can be rewritten as <strong>17 x − x² = 72</strong>. Subtracting 72 from both sides of the equation yields <strong>x² − 17 x + 72 = 0</strong>. Factoring the left-hand side of this equation yields <strong>(x − 8, ) · (x − 9, ) = 0</strong>. Applying the Zero Product Property, it follows that <strong>x − 8 = 0</strong> and <strong>x − 9 = 0</strong>. Solving each equation for x yields <strong>x = 8</strong> and <strong>x = 9</strong> respectively. Note that 8 and 9 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "3b4b8831",
    type: "spr",
    questionHtml:
      "<strong>38 x² = 38 (9)</strong><br>What is the negative solution to the given equation?",
    choices: [],
    correctAnswer: "-3",
    explanation:
      "The correct answer is <strong>−3</strong>. Dividing both sides of the given equation by <strong>38</strong> yields <strong>x² = 9</strong>. Taking the square root of both sides of this equation yields the solutions <strong>x = 3</strong> and <strong>x = −3</strong>. Therefore, the negative solution to the given equation is <strong>−3</strong>.",
    hasFigure: false,
  },
  {
    id: "5ae186b4",
    type: "spr",
    questionHtml:
      "<strong>(−54) / (w) = 6</strong><br>What is the solution to the given equation?",
    choices: [],
    correctAnswer: "-9",
    explanation:
      "The correct answer is <strong>−9</strong>. Since <strong>w</strong> is in the denominator of a fraction in the given equation, <strong>w</strong> can't be equal to <strong>0</strong>. Since <strong>w</strong> isn't equal to <strong>0</strong>, multiplying both sides of the given equation by <strong>w</strong> yields an equivalent equation, <strong>−54 = 6 w</strong>. Dividing both sides of this equation by <strong>6</strong> yields <strong>−9 = w</strong>. Therefore, <strong>−9</strong> is the solution to the given equation.",
    hasFigure: false,
  },
  {
    id: "630897df",
    type: "mcq",
    questionHtml:
      "The speed of sound in dry air, v, can be modeled by the formula <strong>v = 331 . 3 + 0 . 6 0 6 · T</strong>, where T is the temperature in degrees Celsius and v is measured in meters per second. Which of the following correctly expresses T in terms of v ?",
    choices: [
      {
        label: "A",
        text: "<strong>T = the fraction with numerator, v + 0 . 6 0 6, and denominator, 331 . 3</strong>",
      },
      {
        label: "B",
        text: "<strong>T = the fraction with numerator, v − 0 . 6 0 6, and denominator, 331 . 3</strong>",
      },
      {
        label: "C",
        text: "<strong>T = the fraction with numerator, v + 331 . 3, and denominator, 0 . 6 0 6</strong>",
      },
      {
        label: "D",
        text: "<strong>T = the fraction with numerator, v − 331 . 3, and denominator, 0 . 6 0 6</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. To express T in terms of v, subtract 331.3 from both sides of the equation, which gives v – 331.3 = 0.606T. Dividing both sides of the equation by 0.606 gives <strong>the fraction with numerator v − 331 . 3, and denominator 0 . 6 0 6 = T</strong>.Choices A, B, and C are incorrect and are the result of incorrect steps while solving for T.",
    hasFigure: false,
  },
  {
    id: "652054da",
    type: "mcq",
    questionHtml:
      "An oceanographer uses the equation <strong>s = three halves p</strong> to model the speed s, in knots, of an ocean wave, where p represents the period of the wave, in seconds. Which of the following represents the period of the wave in terms of the speed of the wave?",
    choices: [
      { label: "A", text: "<strong>p = two thirds s</strong>" },
      { label: "B", text: "<strong>p = three halves s</strong>" },
      { label: "C", text: "<strong>p = two thirds + s</strong>" },
      { label: "D", text: "<strong>p = three halves + s</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that p represents the period of the ocean wave, so the equation <strong>s = three halves p</strong> can be solved for p to represent the period of the wave in terms of the speed of the wave. Multiplying both sides of the equation by the reciprocal of <strong>three halves</strong> will isolate p. This yields <strong>two thirds s = two thirds · three halves p</strong>, which simplifies to <strong>two thirds s = p</strong>. Therefore, <strong>p = two thirds s</strong>.Choices B, C, and D are incorrect and may result from errors made when rearranging the equation to solve for p.",
    hasFigure: false,
  },
  {
    id: "6e02cd78",
    type: "spr",
    questionHtml:
      "In the xy-plane, what is the y-coordinate of the point of intersection of the graphs of <strong>y = (x − 1, ), ²</strong> and <strong>y = 2 x − 3</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 1. The point of intersection of the graphs of the given equations is the solution to the system of the two equations. Since <strong>y = (x − 1, ), ²</strong> and <strong>y = 2 x − 3</strong>, it follows that <strong>(x − 1, ), ² = 2 x − 3</strong>, or <strong>(x − 1, ) · (x − 1, ) = 2 x − 3</strong>. Applying the distributive property to the left-hand side of this equation yields <strong>x² − 2 x + 1 = 2 x − 3</strong>. Subtracting <strong>2 x</strong> from and adding 3 to both sides of this equation yields <strong>x² − 4 x + 4 = 0</strong>. Factoring the left-hand side of this equation yields <strong>(x − 2, ) · (x − 2, ) = 0</strong>. By the zero product property, if <strong>(x − 2, ) · (x − 2, ) = 0</strong>, it follows that <strong>x − 2 = 0</strong>. Adding 2 to both sides of <strong>x − 2 = 0</strong> yields <strong>x = 2</strong>. Substituting 2 for x in either of the given equations yields <strong>y = 1</strong>. For example, substituting 2 for x in the second given equation yields <strong>y = 2 · 2 − 3</strong>, or <strong>y = 1</strong>. Therefore, the point of intersection of the graphs of the given equations is <strong>2, 1</strong>. The y-coordinate of this point is 1.",
    hasFigure: false,
  },
  {
    id: "717a1964",
    type: "spr",
    questionHtml:
      "<strong>z² + 10 z − 24 = 0</strong><br>What is one of the solutions to the given equation?",
    choices: [],
    correctAnswer: "2, -12",
    explanation:
      "The correct answer is either <strong>2</strong> or <strong>−12</strong>. The left-hand side of the given equation can be rewritten by factoring. The two values that multiply to <strong>−24</strong> and add to <strong>10</strong> are <strong>12</strong> and <strong>−2</strong>. It follows that the given equation can be rewritten as <strong>(z + 12) (z − 2) = 0</strong>. Setting each factor equal to <strong>0</strong> yields two equations: <strong>z + 12 = 0</strong> and <strong>z − 2 = 0</strong>. Subtracting <strong>12</strong> from both sides of the equation <strong>z + 12 = 0</strong> results in <strong>z = −12</strong>. Adding <strong>2</strong> to both sides of the equation <strong>z − 2 = 0</strong> results in <strong>z = 2</strong>. Note that 2 and -12 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "7f81d0c3",
    type: "mcq",
    questionHtml: "What values satisfy the equation above?",
    choices: [
      { label: "A", text: "<strong>x = 1</strong> and <strong>x = 2</strong>" },
      {
        label: "B",
        text: "<strong>x = −one half</strong> and <strong>x = three halves</strong>",
      },
      {
        label: "C",
        text: "<strong>x = the fraction with numerator 1 + the √ 5, and denominator 2</strong> and <strong>x = the fraction with numerator 1 − the √ 5, and denominator 2</strong>",
      },
      {
        label: "D",
        text: "<strong>x = the fraction with numerator − 1 + the √ 5, and denominator 2</strong> and <strong>x = the fraction with numerator − 1 − the √ 5, and denominator 2</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Using the quadratic formula to solve the given expression yields <strong>x = the fraction with numerator negative, (−1, ) + or − the √ (−1, ), ² − 4 · 1 · −1, end root, and denominator, 2 · 1, end fraction, which = the fraction with numerator 1 + or − the √ 5, and denominator 2</strong>. Therefore, <strong>x = the fraction with numerator 1 + the √ 5, and denominator 2</strong> and <strong>x = the fraction with numerator 1 − the √ 5, and denominator 2</strong> satisfy the given equation.Choices A and B are incorrect and may result from incorrectly factoring or incorrectly applying the quadratic formula. Choice D is incorrect and may result from a sign error.",
    hasFigure: false,
  },
  {
    id: "802549ac",
    type: "mcq",
    questionHtml: "Which of the following is a solution to the given equation?",
    choices: [
      { label: "A", text: "<strong>1</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>−2</strong>" },
      { label: "D", text: "<strong>−5</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Applying the distributive property on the left- and right-hand sides of the given equation yields <strong>x² + 2 x + 3 x + 6 = x² − 2 x − 3 x + 6 + 10</strong>, or <strong>x² + 5 x + 6 = x² − 5 x + 16</strong>. Subtracting <strong>x²</strong> from and adding <strong>5 x</strong> to both sides of this equation yields <strong>10 x + 6 = 16</strong>. Subtracting 6 from both sides of this equation and then dividing both sides by 10 yields <strong>x = 1</strong>.Choices B, C, and D are incorrect. Substituting 0, <strong>−2</strong>, or <strong>−5</strong> for x in the given equation will result in a false statement. If <strong>x = 0</strong>, the given equation becomes <strong>6 = 16</strong>; if <strong>x = −2</strong>, the given equation becomes <strong>0 = 30</strong>; and if <strong>x = −5</strong>, the given equation becomes <strong>6 = 66</strong>. Therefore, the values 0, <strong>−2</strong>, and <strong>−5</strong> aren’t solutions to the given equation.",
    hasFigure: false,
  },
  {
    id: "876a731c",
    type: "mcq",
    questionHtml:
      "If (x, y) is a solution of the system of equations above and x > 0, what is the value of xy ?",
    choices: [
      { label: "A", text: "1" },
      { label: "B", text: "2" },
      { label: "C", text: "3" },
      { label: "D", text: "9" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Substituting <strong>x²</strong> for y in the second equation gives <strong>2 · x² + 6 = 2 · (x + 3, )</strong>. This equation can be solved as follows:<br> <strong>2 x² + 6 = 2 x + 6</strong><br> Apply the distributive property.<br> <strong>2 x² + 6 − 2 x − 6 = 0</strong><br> Subtract 2x and 6 from both sides of the equation.<br> <strong>2 x² − 2 x = 0</strong><br> Combine like terms.<br> <strong>2 x · (x − 1, ) = 0</strong><br> Factor both terms on the left side of the equation<br><br> by 2x.<br> Thus, <strong>x = 0</strong> and <strong>x = 1</strong> are the solutions to the system. Since <strong>x > 0</strong>, only <strong>x = 1</strong> needs to be considered. The value of y when <strong>x = 1</strong> is <strong>y = x², which = 1², which = 1</strong>. Therefore, the value of xy is <strong>1 · 1 = 1</strong>.<br>Choices B, C, and D are incorrect and likely result from a computational or conceptual error when solving this system of equations.",
    hasFigure: false,
  },
  {
    id: "87a3de81",
    type: "spr",
    questionHtml:
      "If a is a solution of the equation above and <strong>a > 0</strong>, what is the value of a ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 3. The solution to the given equation can be found by factoring the quadratic expression. The factors can be determined by finding two numbers with a sum of 1 and a product of <strong>−12</strong>. The two numbers that meet these constraints are 4 and <strong>−3</strong>. Therefore, the given equation can be rewritten as <strong>(x + 4, ) · (x − 3, ) = 0</strong>. It follows that the solutions to the equation are <strong>x = −4</strong> or <strong>x = 3</strong>. Since it is given that <strong>a > 0</strong>, a must equal 3.",
    hasFigure: false,
  },
  {
    id: "895628b5",
    type: "mcq",
    questionHtml:
      "<strong>y = (x − 2) (x + 4)</strong><br><strong>y = 6 x − 12</strong><br>Which ordered pair <strong>(x, y)</strong> is the solution to the given system of equations?",
    choices: [
      { label: "A", text: "<strong>(0, 2)</strong>" },
      { label: "B", text: "<strong>(−4, 2)</strong>" },
      { label: "C", text: "<strong>(2, 0)</strong>" },
      { label: "D", text: "<strong>(2 −4)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The second equation in the given system of equations is <strong>y = 6 x − 12</strong>. Substituting <strong>6 x − 12</strong> for <strong>y</strong> in the first equation of the given system yields <strong>6 x − 12 = (x − 2) (x + 4)</strong>. Factoring <strong>6</strong> out of the left-hand side of this equation yields <strong>6 (x − 2) = (x − 2) (x + 4)</strong>. An expression with a factor of the form <strong>(x − a)</strong> is equal to zero when <strong>x = a</strong>. Each side of this equation has a factor of <strong>(x − 2)</strong>, so each side of the equation is equal to zero when <strong>x = 2</strong>. Substituting <strong>2</strong> for <strong>x</strong> into the equation <strong>6 (x − 2) = (x − 2) (x + 4)</strong> yields <strong>6 (2 − 2) = (2 − 2) (2 + 4)</strong>, or <strong>0 = 0</strong>, which is true. Substituting <strong>2</strong> for <strong>x</strong> into the second equation in the given system of equations yields <strong>y = 6 (2) − 12</strong>, or <strong>y = 0</strong>. Therefore, the solution to the system of equations is the ordered pair <strong>(2, 0)</strong>.<br>Choice A is incorrect and may result from switching the order of the solutions for <strong>x</strong> and <strong>y</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "8f65cddc",
    type: "mcq",
    questionHtml:
      "<strong>(1) / (7 b) = (11 x) / (y)</strong><br>The given equation relates the positive numbers <strong>b</strong>, <strong>x</strong>, and <strong>y</strong>. Which equation correctly expresses <strong>x</strong> in terms of <strong>b</strong> and <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>x = (7 b y) / (11)</strong>" },
      { label: "B", text: "<strong>x = y − 77 b</strong>" },
      { label: "C", text: "<strong>x = (y) / (77 b)</strong>" },
      { label: "D", text: "<strong>x = 77 b y</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Multiplying each side of the given equation by <strong>y</strong> yields the equivalent equation <strong>(y) / (7 b) = 11 x</strong>. Dividing each side of this equation by <strong>11</strong> yields <strong>(y) / (77 b) = x</strong>, or <strong>x = (y) / (77 b)</strong>.<br>Choice A is incorrect. This equation is not equivalent to the given equation. <br>Choice B is incorrect. This equation is not equivalent to the given equation. <br>Choice D is incorrect. This equation is not equivalent to the given equation.",
    hasFigure: false,
  },
  {
    id: "911383f2",
    type: "mcq",
    questionHtml: "What is the product of the solutions to the given equation?",
    choices: [
      { label: "A", text: "<strong>8</strong>" },
      { label: "B", text: "<strong>3</strong>" },
      { label: "C", text: "<strong>−3</strong>" },
      { label: "D", text: "<strong>−8</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. By the zero-product property, if <strong>(x − 4, ) · (x + 2, ) · (x − 1, ) = 0</strong>, then <strong>x − 4 = 0</strong>, <strong>x + 2 = 0</strong>, or <strong>x − 1 = 0</strong>. Solving each of these equations for <strong>x</strong> yields <strong>x = 4</strong>, <strong>x = −2</strong>, or <strong>x = 1</strong>. The product of these solutions is <strong>4 · −2 · 1 = −8</strong>.Choice A is incorrect and may result from sign errors made when solving the given equation. Choice B is incorrect and may result from finding the sum, not the product, of the solutions. Choice C is incorrect and may result from finding the sum, not the product, of the solutions in addition to making sign errors when solving the given equation.",
    hasFigure: false,
  },
  {
    id: "928498f3",
    type: "mcq",
    questionHtml: "What are the solutions to the given equation?",
    choices: [
      {
        label: "A",
        text: "<strong>the fraction with numerator − 5 + or − the √ 25 + 168, end root, and denominator 12</strong>",
      },
      {
        label: "B",
        text: "<strong>the fraction with numerator − 6 + or − the √ 25 + 168, end root, and denominator 12</strong>",
      },
      {
        label: "C",
        text: "<strong>the fraction with numerator − 5 + or − the √ 36 − 168, end root, and denominator 12</strong>",
      },
      {
        label: "D",
        text: "<strong>the fraction with numerator − 6 + or − the √ 36 − 168, end root, and denominator 12</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The quadratic formula, <strong>x = the fraction with numerator − b + or − the √, b² − 4 a, c, end root, and denominator 2 a, end fraction</strong>, can be used to find the solutions to an equation in the form <strong>a, x² + b x + c = 0</strong>. In the given equation, <strong>a = 6</strong>, <strong>b = 5</strong>, and <strong>c = −7</strong>. Substituting these values into the quadratic formula gives <strong>the fraction with numerator − 5 + or − the √, 5² − 4 · 6 · 7, end root, and denominator 2 · 6, end fraction</strong>, or <strong>the fraction with numerator − 5 + or − the √, 25 − 168, end root, and denominator 12, end fraction</strong>.Choice B is incorrect and may result from using <strong>the fraction with numerator − a + or − the √, b² − 4 a, c, end root, and denominator 2 a, end fraction</strong> as the quadratic formula. Choice C is incorrect and may result from using<strong>the fraction with numerator − b + or − the √, a, ² + 4 a, c, end root, and denominator 2 a, end fraction</strong> as the quadratic formula. Choice D is incorrect and may result from using <strong>the fraction with numerator − a + or − the √, a, ² + 4 a, c, end root, and denominator 2 a, end fraction</strong> as the quadratic formula.",
    hasFigure: false,
  },
  {
    id: "95ed0b69",
    type: "mcq",
    questionHtml:
      "<strong>p = (k) / (4 j + 9)</strong><br>The given equation relates the distinct positive numbers <strong>p</strong>, <strong>k</strong>, and <strong>j</strong>. Which equation correctly expresses <strong>4 j + 9</strong> in terms of <strong>p</strong> and <strong>k</strong>?",
    choices: [
      { label: "A", text: "<strong>4 j + 9 = (k) / (p)</strong>" },
      { label: "B", text: "<strong>4 j + 9 = k p</strong>" },
      { label: "C", text: "<strong>4 j + 9 = k − p</strong>" },
      { label: "D", text: "<strong>4 j + 9 = (p) / (k)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. To express <strong>4 j + 9</strong> in terms of <strong>p</strong> and <strong>k</strong>, the given equation must be solved for <strong>4 j + 9</strong>. Since it's given that <strong>j</strong> is a positive number, <strong>4 j + 9</strong> is not equal to zero. Therefore, multiplying both sides of the given equation by <strong>4 j + 9</strong> yields the equivalent equation <strong>p (4 j + 9) = k</strong>. Since it's given that <strong>p</strong> is a positive number, <strong>p</strong> is not equal to zero. Therefore, dividing each side of the equation <strong>p (4 j + 9) = k</strong> by <strong>p</strong> yields the equivalent equation <strong>4 j + 9 = (k) / (p)</strong>.<br>Choice B is incorrect. This equation is equivalent to <strong>p = (4 j + 9) / (k)</strong>.<br>Choice C is incorrect. This equation is equivalent to <strong>p = k − 4 j − 9</strong>.<br>Choice D is incorrect. This equation is equivalent to <strong>p = k (4 j + 9)</strong>.",
    hasFigure: false,
  },
  {
    id: "a4f61d75",
    type: "spr",
    questionHtml:
      "In the equation above, a is a constant and <strong>a > 0</strong>. If the equation has two integer solutions, what is a possible value of a ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is either 7, 8, or 13. Since the given equation has two integer solutions, the expression on the left-hand side of this equation can be factored as <strong>(x + c, ) · (x + d, )</strong>, where c and d are also integers. The product of c and d must equal the constant term of the original quadratic expression, which is 12. Additionally, the sum of c and d must be a negative number since it’s given that <strong>a > 0</strong>, but the sign preceding a in the given equation is negative. The possible pairs of values for c and d that satisfy both of these conditions are <strong>−4</strong> and <strong>−3</strong>, <strong>−6</strong> and <strong>−2</strong>, and <strong>−12</strong> and <strong>−1</strong>. Since the value of <strong>−a</strong> is the sum of c and d, the possible values of <strong>−a</strong> are <strong>−4 + −3 = −7</strong>, <strong>−6 + −2 = −8</strong>, and <strong>−12 + −1 = −13</strong>. It follows that the possible values of a are 7, 8, and 13. Note that 7, 8, and 13 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "a5663025",
    type: "mcq",
    questionHtml:
      "A system of equations consists of a quadratic equation and a linear equation. The equations in this system are graphed in the xy-plane above. How many solutions does this system have?",
    choices: [
      { label: "A", text: "0" },
      { label: "B", text: "1" },
      { label: "C", text: "2" },
      { label: "D", text: "3" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The solutions to a system of two equations correspond to points where the graphs of the equations intersect. The given graphs intersect at 2 points; therefore, the system has 2 solutions.Choice A is incorrect because the graphs intersect. Choice B is incorrect because the graphs intersect more than once. Choice D is incorrect. It’s not possible for the graph of a quadratic equation and the graph of a linear equation to intersect more than twice.",
    hasFigure: true,
    figureUrl: "/practice-images/a5663025_img1.png",
  },
  {
    id: "b80d10d7",
    type: "mcq",
    questionHtml: "What is the solution to the equation above?",
    choices: [
      { label: "A", text: "0" },
      { label: "B", text: "2" },
      { label: "C", text: "3" },
      { label: "D", text: "5" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since <strong>the fraction with numerator x + 5, and denominator x + 5, end fraction</strong> is equivalent to 1, the right-hand side of the given equation can be rewritten as <strong>the fraction with numerator x + 5, and denominator x + 5, end fraction − the fraction 1 over, x + 5, end fraction</strong>, or <strong>the fraction with numerator x + 4, and denominator x + 5, end fraction</strong>. Since the left- and right-hand sides of the equation <strong>the fraction with numerator 2 · (x + 1, ), and denominator x + 5, end fraction = the fraction with numerator x + 4, and denominator x + 5, end fraction</strong> have the same denominator, it follows that <strong>2 · (x + 1, ) = x + 4</strong>. Applying the distributive property of multiplication to the expression <strong>2 · (x + 1, )</strong> yields <strong>2 · x + 2 · 1</strong>, or <strong>2 x + 2</strong>. Therefore, <strong>2 x + 2 = x + 4</strong>. Subtracting x and 2 from both sides of this equation yields <strong>x = 2</strong>.Choices A, C, and D are incorrect. If <strong>x = 0</strong>, then <strong>the fraction with numerator 2 · (0 + 1, ), and denominator 0 + 5, end fraction = 1 − the fraction with numerator 1, and denominator 0 + 5, end fraction</strong>. This can be rewritten as <strong>2 over 5 = 4 over 5</strong>, which is a false statement. Therefore, 0 isn’t a solution to the given equation. Substituting 3 and 5 into the given equation yields similarly false statements.",
    hasFigure: false,
  },
  {
    id: "bef4b1c6",
    type: "spr",
    questionHtml:
      "<strong>(55) / (x + 6) = x</strong><br>What is the positive solution to the given equation?",
    choices: [],
    correctAnswer: "5",
    explanation:
      "The correct answer is <strong>5</strong>. Multiplying both sides of the given equation by <strong>x + 6</strong> results in <strong>55 = x (x + 6)</strong>. Applying the distributive property of multiplication to the right-hand side of this equation results in <strong>55 = x² + 6 x</strong>. Subtracting <strong>55</strong> from both sides of this equation results in <strong>0 = x² + 6 x − 55</strong>. The right-hand side of this equation can be rewritten by factoring. The two values that multiply to <strong>−55</strong> and add to <strong>6</strong> are <strong>11</strong> and <strong>−5</strong>. It follows that the equation <strong>0 = x² + 6 x − 55</strong> can be rewritten as <strong>0 = (x + 11) (x − 5)</strong>. Setting each factor equal to <strong>0</strong> yields two equations: <strong>x + 11 = 0</strong> and <strong>x − 5 = 0</strong>. Subtracting <strong>11</strong> from both sides of the equation <strong>x + 11 = 0</strong> results in <strong>x = −11</strong>. Adding <strong>5</strong> to both sides of the equation <strong>x − 5 = 0</strong> results in <strong>x = 5</strong>. Therefore, the positive solution to the given equation is <strong>5</strong>.",
    hasFigure: false,
  },
  {
    id: "c77ef2fb",
    type: "mcq",
    questionHtml:
      "Blood volume,<strong>V sub B</strong>, in a human can be determined using the equation <strong>V sub B = the fraction with numerator V sub P, and denominator 1 − H, end fraction</strong>, where <strong>V sub P</strong> is the plasma volume and H is the hematocrit (the fraction of blood volume that is red blood cells). Which of the following correctly expresses the hematocrit in terms of the blood volume and the plasma volume?",
    choices: [
      {
        label: "A",
        text: "<strong>H = 1 − the fraction V sub P over V sub B</strong>",
      },
      {
        label: "B",
        text: "<strong>H = the fraction V sub B over V sub P</strong>",
      },
      {
        label: "C",
        text: "<strong>H = 1 + the fraction V sub B over V sub P</strong>",
      },
      { label: "D", text: "<strong>H = V sub B − V sub P</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The hematocrit can be expressed in terms of the blood volume and the plasma volume by solving the given equation <strong>V sub B = the fraction with numerator V sub P, and denominator 1 − H, end fraction</strong> for H. Multiplying both sides of this equation by <strong>(1 − H, )</strong> yields <strong>V sub B · (1 − H, ) = V sub P</strong>. Dividing both sides by <strong>V sub B</strong> yields <strong>1 − H = the fraction V sub P, over V sub B</strong>. Subtracting 1 from both sides yields <strong>−H = −1 + the fraction V sub P over V sub B</strong>. Dividing both sides by <strong>−1</strong> yields <strong>H = 1 − the fraction V sub P over V sub B</strong>.Choices B, C, and D are incorrect and may result from errors made when manipulating the equation.",
    hasFigure: false,
  },
  {
    id: "d0a7871e",
    type: "mcq",
    questionHtml:
      "If <strong>the ordered pair x, y</strong> is a solution to the system of equations above, which of the following could be the value of x ?",
    choices: [
      { label: "A", text: "–1" },
      { label: "B", text: "0" },
      { label: "C", text: "2" },
      { label: "D", text: "3" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It is given that y = x + 1 and y = x2 + x. Setting the values for y equal to each other yields x + 1 = x2 + x. Subtracting x from each side of this equation yields x2 = 1. Therefore, x can equal 1 or –1. Of these, only –1 is given as a choice.Choice B is incorrect. If x = 0, then x + 1 = 1, but x2 + x = 02 + 0 = 0 ≠︀ 1. Choice C is incorrect. If x = 2, then x + 1 = 3, but x2 + x = 22 + 2 = 6 ≠︀ 3. Choice D is incorrect. If x = 3, then x + 1 = 4, but x2 + x = 32 + 3 = 12 ≠︀ 4.",
    hasFigure: false,
  },
  {
    id: "da602115",
    type: "spr",
    questionHtml:
      "If <strong>|4 x − 4| = 112</strong>, what is the positive value of <strong>x − 1</strong>?",
    choices: [],
    correctAnswer: "28",
    explanation:
      "The correct answer is <strong>28</strong>. The given absolute value equation can be rewritten as two linear equations: <strong>4 x − 4 = 112</strong> and <strong>− (4 x − 4) = 112</strong>, or <strong>4 x − 4 = −112</strong>. Adding <strong>4</strong> to both sides of the equation <strong>4 x − 4 = 112</strong> results in <strong>4 x = 116</strong>. Dividing both sides of this equation by <strong>4</strong> results in <strong>x = 29</strong>. Adding <strong>4</strong> to both sides of the equation <strong>4 x − 4 = −112</strong> results in <strong>4 x = −108</strong>. Dividing both sides of this equation by <strong>4</strong> results in <strong>x = −27</strong>. Therefore, the two values of <strong>x − 1</strong> are <strong>29 − 1</strong>, or <strong>28</strong>, and <strong>−27 − 1</strong>, or <strong>−28</strong>. Thus, the positive value of <strong>x − 1</strong> is <strong>28</strong>.<br>Alternate approach: The given equation can be rewritten as <strong>|4 (x − 1)| = 112</strong>, which is equivalent to <strong>4 |x − 1| = 112</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>|x − 1| = 28</strong>. This equation can be rewritten as two linear equations: <strong>x − 1 = 28</strong> and <strong>− (x − 1) = 28</strong>, or <strong>x − 1 = −28</strong>. Therefore, the positive value of <strong>x − 1</strong> is <strong>28</strong>.",
    hasFigure: false,
  },
  {
    id: "e8779461",
    type: "spr",
    questionHtml:
      "<strong>y = x² + 14 x + 48</strong><br><strong>x + 8 = 11</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [],
    correctAnswer: "99",
    explanation:
      "The correct answer is <strong>99</strong>. In the given system of equations, the second equation is <strong>x + 8 = 11</strong>. Subtracting <strong>8</strong> from both sides of this equation yields <strong>x = 3</strong>. In the given system of equations, the first equation is <strong>y = x² + 14 x + 48</strong>. Substituting <strong>3</strong> for <strong>x</strong> in this equation yields <strong>y = (3)² + 14 (3) + 48</strong>, or <strong>y = 99</strong>. Therefore, the solution to the given system of equations is <strong>(x, y) = (3, 99)</strong>. Thus, the value of <strong>y</strong> is <strong>99</strong>.",
    hasFigure: false,
  },
  {
    id: "f5247e52",
    type: "mcq",
    questionHtml:
      "In the equation above, a and c are positive constants. How many times does the graph of the equation above intersect the graph of the equation <strong>y = a + c</strong> in the xy-plane?",
    choices: [
      { label: "A", text: "Zero" },
      { label: "B", text: "One" },
      { label: "C", text: "Two" },
      { label: "D", text: "More than two" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It is given that the constants a and c are both positive; therefore, the graph of the given quadratic equation is a parabola that opens up with a vertex on the y-axis at a point below the x-axis. The graph of the second equation provided is a horizontal line that lies above the x-axis. A horizontal line above the x-axis will intersect a parabola that opens up and has a vertex below the x-axis in exactly two points.Choices A, B, and D are incorrect and are the result of not understanding the relationships of the graphs of the two equations given. Choice A is incorrect because the two graphs intersect. Choice B is incorrect because in order for there to be only one intersection point, the horizontal line would have to intersect the parabola at the vertex, but the vertex is below the x-axis and the line is above the x-axis. Choice D is incorrect because a line cannot intersect a parabola in more than two points.",
    hasFigure: false,
  },
  {
    id: "fcdf87b7",
    type: "spr",
    questionHtml:
      "If the ordered pair <strong>x, y</strong> satisfies the system of equations above, what is one possible value of x ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is either 0 or 3. For an ordered pair to satisfy a system of equations, both the x- and y-values of the ordered pair must satisfy each equation in the system. Both expressions on the right-hand side of the given equations are equal to y, therefore it follows that both expressions on the right-hand side of the equations are equal to each other: <strong>x² − 4 x + 4 = 4 − x</strong>. This equation can be rewritten as <strong>x² − 3 x = 0</strong>, and then through factoring, the equation becomes <strong>x · (x − 3, ) = 0</strong>. Because the product of the two factors is equal to 0, it can be concluded that either <strong>x = 0</strong> or <strong>x − 3 = 0</strong>, or rather, <strong>x = 0</strong> or <strong>x = 3</strong>. Note that 0 and 3 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "ff2c1431",
    type: "mcq",
    questionHtml:
      "<strong>7 m = 5 (n + p)</strong><br>The given equation relates the positive numbers <strong>m</strong>, <strong>n</strong>, and <strong>p</strong>. Which equation correctly gives <strong>n</strong> in terms of <strong>m</strong> and <strong>p</strong>?",
    choices: [
      { label: "A", text: "<strong>n = (5 p) / (7 m)</strong>" },
      { label: "B", text: "<strong>n = (7 m) / (5) − p</strong>" },
      { label: "C", text: "<strong>n = 5 (7 m) + p</strong>" },
      { label: "D", text: "<strong>n = 7 m − 5 − p</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that the equation <strong>7 m = 5 (n + p)</strong> relates the positive numbers <strong>m</strong>, <strong>n</strong>, and <strong>p</strong>. Dividing both sides of the given equation by <strong>5</strong> yields <strong>(7 m) / (5) = n + p</strong>. Subtracting <strong>p</strong> from both sides of this equation yields <strong>(7 m) / (5) − p = n</strong>, or <strong>n = (7 m) / (5) − p</strong>. It follows that the equation <strong>n = (7 m) / (5) − p</strong> correctly gives <strong>n</strong> in terms of <strong>m</strong> and <strong>p</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];

export const NONLINEAR_EQ_HARD: PracticeQuestion[] = [
  {
    id: "03ff48d2",
    type: "spr",
    questionHtml:
      "<strong>x (k x − 56) = −16</strong><br>In the given equation, <strong>k</strong> is an integer constant. If the equation has no real solution, what is the least possible value of <strong>k</strong>?",
    choices: [],
    correctAnswer: "50",
    explanation:
      "The correct answer is <strong>50</strong>. An equation of the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, has no real solutions if and only if its discriminant, <strong>b² − 4 a c</strong>, is negative. Applying the distributive property to the left-hand side of the equation <strong>x (k x − 56) = −16</strong> yields <strong>k x² − 56 x = −16</strong>. Adding <strong>16</strong> to each side of this equation yields <strong>k x² − 56 x + 16 = 0</strong>. Substituting <strong>k</strong> for <strong>a</strong>, <strong>−56</strong> for <strong>b</strong>, and <strong>16</strong> for <strong>c</strong> in <strong>b² − 4 a c</strong> yields a discriminant of <strong>(−56)² − 4 (k) (16)</strong>, or <strong>3, 136 − 64 k</strong>. If the given equation has no real solution, it follows that the value of <strong>3, 136 − 64 k</strong> must be negative. Therefore, <strong>3, 136 − 64 k < 0</strong>. Adding <strong>64 k</strong> to both sides of this inequality yields <strong>3, 136 < 64 k</strong>. Dividing both sides of this inequality by <strong>64</strong> yields <strong>49 < k</strong>, or <strong>k > 49</strong>. Since it's given that <strong>k</strong> is an integer, the least possible value of <strong>k</strong> is <strong>50</strong>.",
    hasFigure: false,
  },
  {
    id: "104bff62",
    type: "mcq",
    questionHtml:
      "<strong>(x²) / (√(x² − c²)) = (c²) / (√(x² − c²)) + 39</strong><br>In the given equation, <strong>c</strong> is a positive constant. Which of the following is one of the solutions to the given equation?",
    choices: [
      { label: "A", text: "<strong>−c</strong>" },
      { label: "B", text: "<strong>− c² − 39²</strong>" },
      { label: "C", text: "<strong>− √(39² − c²)</strong>" },
      { label: "D", text: "<strong>− √(c² + 39²)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. If <strong>x² − c² < or = 0</strong>, then neither side of the given equation is defined and there can be no solution. Therefore, <strong>x² − c² > 0</strong>. Subtracting <strong>(c²) / (√(x² − c²))</strong> from both sides of the given equation yields <strong>(x²) / (√(x² − c²)) − (c²) / (√(x² − c²)) = 39</strong>, or <strong>(x² − c²) / (√(x² − c²)) = 39</strong>. Squaring both sides of this equation yields <strong>((x² − c²) / (√(x² − c²)))² = 39²</strong>, or <strong>((x² − c²) (x² − c²)) / (x² − c²) = 39²</strong>. Since <strong>x² − c²</strong> is positive and, therefore, nonzero, the expression <strong>(x² − c²) / (x² − c²)</strong> is defined and equivalent to <strong>1</strong>. It follows that the equation <strong>((x² − c²) (x² − c²)) / (x² − c²) = 39²</strong> can be rewritten as <strong>((x² − c²) / (x² − c²)) (x² − c²) = 39²</strong>, or <strong>(1) (x² − c²) = 39²</strong>, which is equivalent to <strong>x² − c² = 39²</strong>. Adding <strong>c²</strong> to both sides of this equation yields <strong>x² = c² + 39²</strong>. Taking the square root of both sides of this equation yields two solutions: <strong>x = √(c² + 39²)</strong> and <strong>x = − √(c² + 39²)</strong>. Therefore, of the given choices, <strong>− √(c² + 39²)</strong> is one of the solutions to the given equation.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "158591f0",
    type: "spr",
    questionHtml:
      "<strong>x (x + 1) − 56 = 4 x (x − 7)</strong><br>What is the sum of the solutions to the given equation?",
    choices: [],
    correctAnswer: "29/3, 9.666, 9.667",
    explanation:
      "The correct answer is <strong>(29) / (3)</strong>. Applying the distributive property to the left-hand side of the given equation, <strong>x (x + 1) − 56</strong>, yields <strong>x² + x − 56</strong>. Applying the distributive property to the right-hand side of the given equation, <strong>4 x (x − 7)</strong>, yields <strong>4 x² − 28 x</strong>. Thus, the equation becomes <strong>x² + x − 56 = 4 x² − 28 x</strong>. Combining like terms on the left- and right-hand sides of this equation yields <strong>0 = (4 x² − x²) + (−28 x − x) + 56</strong>, or <strong>3 x² − 29 x + 56 = 0</strong>. For a quadratic equation in the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, the quadratic formula gives the solutions to the equation in the form <strong>x = ((−b + or − √(b² − 4 a c))) / (2 a)</strong>. Substituting <strong>3</strong> for <strong>a</strong>, <strong>−29</strong> for <strong>b</strong>, and <strong>56</strong> for <strong>c</strong> from the equation <strong>3 x² − 29 x + 56 = 0</strong> into the quadratic formula yields <strong>x = ((29 + or − √((−29)² − 4 (3) (56)))) / (2 (3))</strong>, or <strong>x = (29) / (6) + or − (13) / (6)</strong>. It follows that the solutions to the given equation are <strong>(29) / (6) + (13) / (6)</strong> and <strong>(29) / (6) − (13) / (6)</strong>. Adding these two solutions gives the sum of the solutions: <strong>(29) / (6) + (13) / (6) + (29) / (6) − (13) / (6)</strong>, which is equivalent to <strong>(29) / (6) + (29) / (6)</strong>, or <strong>(29) / (3)</strong>. Note that 29/3, 9.666, and 9.667 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "1697ffcf",
    type: "spr",
    questionHtml:
      "In the xy-plane, the graph of <strong>y = 3 x² − 14 x</strong> intersects the graph of <strong>y = x</strong> at the points <strong>with coordinates zero, zero</strong> and <strong>a, , a</strong>. What is the value of a ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 5. The intersection points of the graphs of <strong>y = 3 x² − 14 x</strong> and <strong>y = x</strong> can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives <strong>x = 3 x² − 14 x</strong>. Subtracting x from both sides of the equation gives <strong>0 = 3 x² − 15 x</strong>. Factoring <strong>3 x</strong> out of each term on the left-hand side of the equation gives <strong>0 = 3 x · (x − 5, )</strong>. Therefore, the possible values for x are 0 and 5. Since <strong>y = x</strong>, the two intersection points are <strong>0, 0</strong> and <strong>5, 5</strong>. Therefore, <strong>a = 5</strong>.",
    hasFigure: false,
  },
  {
    id: "17d0e87d",
    type: "mcq",
    questionHtml:
      "<strong>(14 x) / (7 y) = 2 √(w + 19)</strong><br>The given equation relates the distinct positive real numbers <strong>w</strong>, <strong>x</strong>, and <strong>y</strong>. Which equation correctly expresses <strong>w</strong> in terms of <strong>x</strong> and <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>w = √((x) / (y)) − 19</strong>" },
      { label: "B", text: "<strong>w = √((28 x) / (14 y)) − 19</strong>" },
      { label: "C", text: "<strong>w = ((x) / (y))² − 19</strong>" },
      { label: "D", text: "<strong>w = ((28 x) / (14 y))² − 19</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Dividing each side of the given equation by <strong>2</strong> yields <strong>(14 x) / (14 y) = (2 √(w + 19)) / (2)</strong>, or <strong>(x) / (y) = √(w + 19)</strong>. Because it's given that each of the variables is positive, squaring each side of this equation yields the equivalent equation <strong>((x) / (y))² = w + 19</strong>. Subtracting <strong>19</strong> from each side of this equation yields <strong>((x) / (y))² − 19 = w</strong>, or <strong>w = ((x) / (y))² − 19</strong>.<br>Choice A is incorrect. This equation isn't equivalent to the given equation. <br>Choice B is incorrect. This equation isn't equivalent to the given equation. <br>Choice D is incorrect. This equation isn't equivalent to the given equation.",
    hasFigure: false,
  },
  {
    id: "1ce9ffcd",
    type: "mcq",
    questionHtml:
      "<strong>− 9 x² + 30 x + c = 0</strong><br>In the given equation, <strong>c</strong> is a constant. The equation has exactly one solution. What is the value of <strong>c</strong>?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>−25</strong>" },
      { label: "D", text: "<strong>−53</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that the equation <strong>− 9 x² + 30 x + c = 0</strong> has exactly one solution. A quadratic equation of the form <strong>a x² + b x + c = 0</strong> has exactly one solution if and only if its discriminant, <strong>− 4 a c + b²</strong>, is equal to zero. It follows that for the given equation, <strong>a = −9</strong> and <strong>b = 30</strong>. Substituting <strong>−9</strong> for <strong>a</strong> and <strong>30</strong> for <strong>b</strong> into <strong>b² − 4 a c</strong>  yields <strong>30² − 4 (−9) (c)</strong>, or <strong>900 + 36 c</strong>. Since the discriminant must equal zero, <strong>900 + 36 c = 0</strong>. Subtracting <strong>36 c</strong> from both sides of this equation yields <strong>900 = − 36 c</strong>. Dividing each side of this equation by <strong>−36</strong> yields <strong>−25 = c</strong>. Therefore, the value of <strong>c</strong> is <strong>−25</strong>. <br>Choice A is incorrect. If the value of <strong>c</strong> is <strong>3</strong>, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.<br>Choice B is incorrect. If the value of <strong>c</strong> is <strong>0</strong>, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.<br>Choice D is incorrect. If the value of <strong>c</strong> is <strong>−53</strong>, this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.",
    hasFigure: false,
  },
  {
    id: "1fe32f7d",
    type: "spr",
    questionHtml:
      "<strong>− x² + b x − 676 = 0</strong><br>In the given equation, <strong>b</strong> is a positive integer. The equation has no real solution. What is the greatest possible value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "51",
    explanation:
      "The correct answer is <strong>51</strong>. A quadratic equation of the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, has no real solution if and only if its discriminant, <strong>− 4 a c + b²</strong>, is negative. In the given equation, <strong>a = −1</strong> and <strong>c = −676</strong>. Substituting <strong>−1</strong> for <strong>a</strong> and <strong>−676</strong> for <strong>c</strong> in this expression yields a discriminant of <strong>b² − 4 (−1) (−676)</strong>, or <strong>b² − 2, 704</strong>. Since this value must be negative, <strong>b² − 2, 704 < 0</strong>, or <strong>b² < 2, 704</strong>. Taking the positive square root of each side of this inequality yields <strong>b < 52</strong>. Since <strong>b</strong> is a positive integer, and the greatest integer less than <strong>52</strong> is <strong>51</strong>, the greatest possible value of <strong>b</strong> is <strong>51</strong>.",
    hasFigure: false,
  },
  {
    id: "2c05d312",
    type: "mcq",
    questionHtml:
      "<strong>57 x² + (57 b + a) x + a b = 0</strong><br>In the given equation, <strong>a</strong> and <strong>b</strong> are positive constants. The product of the solutions to the given equation is <strong>k a b</strong>, where <strong>k</strong> is a constant. What is the value of <strong>k</strong>?",
    choices: [
      { label: "A", text: "<strong>one fifty seventh</strong>" },
      { label: "B", text: "<strong>one nineteenth</strong>" },
      { label: "C", text: "<strong>1</strong>" },
      { label: "D", text: "<strong>57</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The left-hand side of the given equation is the expression <strong>57 x² + (57 b + a) x + a b</strong>. Applying the distributive property to this expression yields <strong>57 x² + 57 b x + a x + a b</strong>. Since the first two terms of this expression have a common factor of <strong>57 x</strong> and the last two terms of this expression have a common factor of <strong>a</strong>, this expression can be rewritten as <strong>57 x (x + b) + a (x + b)</strong>. Since the two terms of this expression have a common factor of <strong>(x + b)</strong>, it can be rewritten as <strong>(x + b) (57 x + a)</strong>. Therefore, the given equation can be rewritten as <strong>(x + b) (57 x + a) = 0</strong>. By the zero product property, it follows that <strong>x + b = 0</strong> or <strong>57 x + a = 0</strong>. Subtracting <strong>b</strong> from both sides of the equation <strong>x + b = 0</strong> yields <strong>x = −b</strong>. Subtracting <strong>a</strong> from both sides of the equation <strong>57 x + a = 0</strong> yields <strong>57 x = −a</strong>. Dividing both sides of this equation by <strong>57</strong> yields <strong>x = (−a) / (57)</strong>. Therefore, the solutions to the given equation are <strong>−b</strong> and <strong>(−a) / (57)</strong>. It follows that the product of the solutions of the given equation is <strong>(−b) ((−a) / (57))</strong>, or <strong>(a b) / (57)</strong>. It’s given that the product of the solutions of the given equation is <strong>k a b</strong>. It follows that <strong>(a b) / (57) = k a b</strong>, which can also be written as <strong>a b (one fifty seventh) = a b (k)</strong>. It’s given that <strong>a</strong> and <strong>b</strong> are positive constants. Therefore, dividing both sides of the equation <strong>a b (one fifty seventh) = a b (k)</strong> by <strong>a b</strong> yields <strong>one fifty seventh = k</strong>. Thus, the value of <strong>k</strong> is <strong>one fifty seventh</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "2c5c22d0",
    type: "mcq",
    questionHtml:
      "How many solutions are there to the system of equations above?",
    choices: [
      { label: "A", text: "There are exactly 4 solutions." },
      { label: "B", text: "There are exactly 2 solutions." },
      { label: "C", text: "There is exactly 1 solution." },
      { label: "D", text: "There are no solutions." },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The second equation of the system can be rewritten as <strong>y = 5 x − 8</strong>. Substituting <strong>5 x − 8</strong> for y in the first equation gives <strong>5 x − 8 = x² + 3 x − 7</strong>. This equation can be solved as shown below:<br> <strong>x² + 3 x − 7 − 5 x + 8 = 0</strong><br><br> <strong>x² − 2 x + 1 = 0</strong><br><br> <strong>(x − 1, ), ² = 0</strong><br><br> <strong>x = 1</strong><br>Substituting 1 for x in the equation <strong>y = 5 x − 8</strong> gives <strong>y = −3</strong>. Therefore, <strong>the ordered pair 1 −3</strong> is the only solution to the system of equations.<br>Choice A is incorrect. In the xy-plane, a parabola and a line can intersect at no more than two points. Since the graph of the first equation is a parabola and the graph of the second equation is a line, the system cannot have more than 2 solutions. Choice B is incorrect. There is a single ordered pair <strong>x, y</strong> that satisfies both equations of the system. Choice D is incorrect because the ordered pair <strong>1 −3</strong> satisfies both equations of the system.",
    hasFigure: false,
  },
  {
    id: "2cd6b22d",
    type: "mcq",
    questionHtml:
      "<strong>5 x² + 10 x + 16 = 0</strong><br>How many distinct real solutions does the given equation have?",
    choices: [
      { label: "A", text: "Exactly one" },
      { label: "B", text: "Exactly two" },
      { label: "C", text: "Infinitely many" },
      { label: "D", text: "Zero" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The number of solutions of a quadratic equation of the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, can be determined by the value of the discriminant, <strong>b² − 4 a c</strong>. If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, <strong>5 x² + 10 x + 16 = 0</strong>, <strong>a = 5</strong>, <strong>b = 10</strong>, and <strong>c = 16</strong>. Substituting these values for <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> in <strong>b² − 4 a c</strong> yields <strong>(10)² − 4 (5) (16)</strong>, or <strong>−220</strong>. Since the value of its discriminant is negative, the given equation has zero real solutions. Therefore, the number of distinct real solutions the given equation has is zero.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "30281058",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the graph of <strong>y = x² − 9</strong> intersects line p at <strong>1, a</strong> and <strong>5, b</strong>, where a and b are constants. What is the slope of line p ?",
    choices: [
      { label: "A", text: "<strong>6</strong>" },
      { label: "B", text: "<strong>2</strong>" },
      { label: "C", text: "<strong>−2</strong>" },
      { label: "D", text: "<strong>−6</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the graph of <strong>y = x² − 9</strong> and line p intersect at <strong>1, a</strong> and <strong>5, b</strong>. Therefore, the value of y when <strong>x = 1</strong> is the value of a, and the value of y when <strong>x = 5</strong> is the value of b. Substituting 1 for x in the given equation yields <strong>y = 1² − 9</strong>, or <strong>y = −8</strong>. Similarly, substituting 5 for x in the given equation yields <strong>y = 5² − 9</strong>, or <strong>y = 16</strong>. Therefore, the intersection points are <strong>1 −8</strong> and <strong>5, 16</strong>. The slope of line p is the ratio of the change in y to the change in x between these two points: <strong>the fraction with numerator 16 − −8, and denominator 5 − 1, end fraction = the fraction 24 over 4</strong>, or 6.Choices B, C, and D are incorrect and may result from conceptual or calculation errors in determining the values of a, b, or the slope of line p.",
    hasFigure: false,
  },
  {
    id: "3a9d60b2",
    type: "spr",
    questionHtml:
      "<strong>2 |4 − x| + 3 |4 − x| = 25</strong><br>What is the positive solution to the given equation?",
    choices: [],
    correctAnswer: "9",
    explanation:
      "The correct answer is <strong>9</strong>. The given equation can be rewritten as <strong>5 |4 − x| = 25</strong>. Dividing each side of this equation by <strong>5</strong> yields <strong>|4 − x| = 5</strong>. By the definition of absolute value, if <strong>|4 − x| = 5</strong>, then <strong>4 − x = 5</strong> or <strong>4 − x = −5</strong>. Subtracting <strong>4</strong> from each side of the equation <strong>4 − x = 5</strong> yields <strong>−x = 1</strong>. Dividing each side of this equation by <strong>−1</strong> yields <strong>x = −1</strong>. Similarly, subtracting <strong>4</strong> from each side of the equation <strong>4 − x = −5</strong> yields <strong>−x = −9</strong>. Dividing each side of this equation by <strong>−1</strong> yields <strong>x = 9</strong>. Therefore, since the two solutions to the given equation are <strong>−1</strong> and <strong>9</strong>, the positive solution to the given equation is <strong>9</strong>.",
    hasFigure: false,
  },
  {
    id: "3d12b1e0",
    type: "spr",
    questionHtml:
      "<strong>− 16 x² − 8 x + c = 0</strong><br>In the given equation, <strong>c</strong> is a constant. The equation has exactly one solution. What is the value of <strong>c</strong>?",
    choices: [],
    correctAnswer: "-1",
    explanation:
      "The correct answer is <strong>−1</strong>. A quadratic equation in the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, has exactly one solution when its discriminant, <strong>b² − 4 a c</strong>, is equal to <strong>0</strong>. In the given equation, <strong>− 16 x² − 8 x + c = 0</strong>, <strong>a = −16</strong> and <strong>b = −8</strong>. Substituting <strong>−16</strong> for <strong>a</strong> and <strong>−8</strong> for <strong>b</strong> in <strong>b² − 4 a c</strong> yields <strong>(−8)² − 4 (−16) (c)</strong>, or <strong>64 + 64 c</strong>. Since the given equation has exactly one solution, <strong>64 + 64 c = 0</strong>. Subtracting <strong>64</strong> from both sides of this equation yields <strong>64 c = −64</strong>. Dividing both sides of this equation by <strong>64</strong> yields <strong>c = −1</strong>. Therefore, the value of <strong>c</strong> is <strong>−1</strong>.",
    hasFigure: false,
  },
  {
    id: "4661e2a9",
    type: "mcq",
    questionHtml:
      "Which ordered pair is a solution to the system of equations above?",
    choices: [
      { label: "A", text: "<strong>1 + the √ 3, , , the √ 3</strong>" },
      { label: "B", text: "<strong>the √ 3, , , the − of the √ 3</strong>" },
      { label: "C", text: "<strong>1 + the √ 5, , , the √ 5</strong>" },
      { label: "D", text: "<strong>the √ 5, , −1 + the √ 5</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The solution to the given system of equations can be found by solving the first equation for x, which gives <strong>x = y + 1</strong>, and substituting that value of x into the second equation which gives <strong>y + 1 + y = (y + 1, ), ² − 3</strong>. Rewriting this equation by adding like terms and expanding <strong>(y + 1, ), ²</strong> gives <strong>2 y + 1 = y² + 2 y − 2</strong>. Subtracting <strong>2 y</strong> from both sides of this equation gives <strong>1 = y² − 2</strong>. Adding to 2 to both sides of this equation gives <strong>3 = y²</strong>. Therefore, it follows that <strong>y = + or − the √ 3</strong>. Substituting <strong>the √ 3</strong> for y in the first equation yields <strong>x − the √ 3 = 1</strong>. Adding <strong>the √ 3</strong> to both sides of this equation yields <strong>x = 1 + the √ 3</strong>. Therefore, the ordered pair <strong>1 + the √ 3, end root, , the √ 3</strong> is a solution to the given system of equations.Choice B is incorrect. Substituting <strong>the √ 3</strong> for x and <strong>the − of the √ 3</strong> for y in the first equation yields <strong>the √ 3, end root − the − of the √ 3 = 1</strong>, or <strong>2 · the √ 3 = 1</strong>, which isn’t a true statement. Choice C is incorrect. Substituting <strong>1 + the √ 5</strong> for x and <strong>the √ 5</strong> for y in the second equation yields <strong>(1 + the √ 5, ) + the √ 5 = (1 + the √ 5, ), ² − 3</strong>, or <strong>1 + 2 · the √ 5 = 2 · the √ 5 + 3</strong>, which isn’t a true statement. Choice D is incorrect. Substituting <strong>the √ 5</strong> for x and <strong>(−1 + the √ 5, )</strong> for y in the second equation yields <strong>the √ 5 + (−1 + the √ 5, ) = the √ 5, ² − 3</strong>, or <strong>2 · the √ 5 − 1 = 2</strong>, which isn’t a true statement.",
    hasFigure: false,
  },
  {
    id: "58b109d4",
    type: "spr",
    questionHtml:
      "<strong>x² + y + 7 = 7</strong><br><strong>20 x + 100 − y = 0</strong><br>The solution to the given system of equations is <strong>(x, y)</strong>. What is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "-10",
    explanation:
      "The correct answer is <strong>−10</strong>. Adding <strong>y</strong> to both sides of the second equation in the given system yields <strong>20 x + 100 = y</strong>. Substituting <strong>20 x + 100</strong> for <strong>y</strong> in the first equation in the given system yields <strong>x² + 20 x + 100 + 7 = 7</strong>. Subtracting <strong>7</strong> from both sides of this equation yields <strong>x² + 20 x + 100 = 0</strong>. Factoring the left-hand side of this equation yields <strong>(x + 10) (x + 10) = 0</strong>, or <strong>(x + 10)² = 0</strong>. Taking the square root of both sides of this equation yields <strong>x + 10 = 0</strong>. Subtracting <strong>10</strong> from both sides of this equation yields <strong>x = −10</strong>. Therefore, the value of <strong>x</strong> is <strong>−10</strong>.",
    hasFigure: false,
  },
  {
    id: "5910bfff",
    type: "mcq",
    questionHtml:
      "<strong>D = T − the fraction 9 over 25, end fraction · (100 − H, )</strong>The formula above can be used to approximate the dew point D, in degrees Fahrenheit, given the temperature T, in degrees Fahrenheit, and the relative humidity of H percent, where <strong>H > 50</strong>. Which of the following expresses the relative humidity in terms of the temperature and the dew point?",
    choices: [
      {
        label: "A",
        text: "<strong>H = the fraction 25 over 9, end fraction · (D − T, ) + 100</strong>",
      },
      {
        label: "B",
        text: "<strong>H = the fraction 25 over 9, end fraction · (D − T, ) − 100</strong>",
      },
      {
        label: "C",
        text: "<strong>H = the fraction 25 over 9, end fraction · (D + T, ) + 100</strong>",
      },
      {
        label: "D",
        text: "<strong>H = the fraction 25 over 9, end fraction · (D + T, ) − 100</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that <strong>D = T − the fraction 9 over 25, end fraction · (100 − H, )</strong>. Solving this formula for H expresses the relative humidity in terms of the temperature and the dew point. Subtracting T from both sides of this equation yields <strong>D − T = the − of the fraction 9 over 25, end fraction · (100 − H, )</strong>. Multiplying both sides by <strong>the − of the fraction 25 over 9</strong> yields <strong>the − of the fraction 25 over 9, end fraction · (D − T, ) = 100 − H</strong>. Subtracting 100 from both sides yields <strong>the − of the fraction 25 over 9, end fraction · (D − T, ) − 100 = −H</strong>. Multiplying both sides by <strong>−1</strong> results in the formula <strong>the fraction 25 over 9, end fraction · (D − T, ) + 100 = H.</strong>.Choices B, C, and D are incorrect and may result from errors made when rewriting the given formula.",
    hasFigure: false,
  },
  {
    id: "5edc8c98",
    type: "spr",
    questionHtml:
      "<strong>64 x² − (16 a + 4 b) x + a b = 0</strong><br>In the given equation, <strong>a</strong> and <strong>b</strong> are positive constants. The sum of the solutions to the given equation is <strong>k (4 a + b)</strong>, where <strong>k</strong> is a constant. What is the value of <strong>k</strong>?",
    choices: [],
    correctAnswer: ".0625, 1/16",
    explanation:
      "The correct answer is <strong>one sixteenth</strong>. Let <strong>p</strong> and <strong>q</strong> represent the solutions to the given equation. Then, the given equation can be rewritten as <strong>64 (x − p) (x − q) = 0</strong>, or <strong>64 x² − 64 (p + q) + p q = 0</strong>. Since this equation is equivalent to the given equation, it follows that <strong>− (16 a + 4 b) = − 64 (p + q)</strong>. Dividing both sides of this equation by <strong>−64</strong> yields <strong>(16 a + 4 b) / (64) = p + q</strong>, or <strong>one sixteenth (4 a + b) = p + q</strong>. Therefore, the sum of the solutions to the given equation, <strong>p + q</strong>, is equal to <strong>one sixteenth (4 a + b)</strong>. Since it's given that the sum of the solutions to the given equation is <strong>k (4 a + b)</strong>, where <strong>k</strong> is a constant, it follows that <strong>k = one sixteenth</strong>. Note that 1/16, .0625, 0.062, and 0.063 are examples of ways to enter a correct answer.<br>Alternate approach: The given equation can be rewritten as <strong>64 x² − 4 (4 a + b) x + a b = 0</strong>, where <strong>a</strong> and <strong>b</strong> are positive constants. Dividing both sides of this equation by <strong>4</strong> yields <strong>16 x² − (4 a + b) x + (a b) / (4) = 0</strong>. The solutions for a quadratic equation in the form <strong>A x² + B x + C = 0</strong>, where <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> are constants, can be calculated using the quadratic formula, <strong>x = (−B + √(B² − 4 A C)) / (2 A)</strong> and <strong>x = (−B − √(B² − 4 A C)) / (2 A)</strong>. It follows that the sum of the solutions to a quadratic equation in the form <strong>A x² + B x + C = 0</strong> is <strong>(−B + √(B² − 4 A C)) / (2 A) + (−B − √(B² − 4 A C)) / (2 A)</strong>, which can be rewritten as <strong>(−B + −B + √(B² − 4 A C) − √(B² − 4 A C)) / (2 A)</strong>, which is equivalent to <strong>(−2 B) / (2 A)</strong>, or <strong>− (B) / (A)</strong>. In the equation <strong>16 x² − (4 a + b) x + (a b) / (4) = 0</strong>, <strong>A = 16</strong>, <strong>B = − (4 a + b)</strong>, and <strong>C = (a b) / (4)</strong>. Substituting <strong>16</strong> for <strong>A</strong> and <strong>− (4 a + b)</strong> for <strong>B</strong> in <strong>− (B) / (A)</strong> yields <strong>− (−(4 a + b)) / (16)</strong>, which can be rewritten as <strong>one sixteenth (4 a + b)</strong>. Thus, the sum of the solutions to the given equation is <strong>one sixteenth (4 a + b)</strong>. Since it's given that the sum of the solutions to the given equation is <strong>k (4 a + b)</strong>, where <strong>k</strong> is a constant, it follows that <strong>k = one sixteenth</strong>.",
    hasFigure: false,
  },
  {
    id: "6011a3f8",
    type: "mcq",
    questionHtml:
      "<strong>64 x² + b x + 25 = 0</strong><br>In the given equation, <strong>b</strong> is a constant. For which of the following values of <strong>b</strong> will the equation have more than one real solution?",
    choices: [
      { label: "A", text: "<strong>−91</strong>" },
      { label: "B", text: "<strong>−80</strong>" },
      { label: "C", text: "<strong>5</strong>" },
      { label: "D", text: "<strong>40</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. A quadratic equation of the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, has either no real solutions, exactly one real solution, or exactly two real solutions. That is, for the given equation to have more than one real solution, it must have exactly two real solutions. When the value of the discriminant, or <strong>b² − 4 a c</strong>, is greater than 0, the given equation has exactly two real solutions. In the given equation, <strong>64 x² + b x + 25 = 0</strong>, <strong>a = 64</strong> and <strong>c = 25</strong>. Therefore, the given equation has exactly two real solutions when <strong>(b)² − 4 (64) (25) > 0</strong>, or <strong>b² − 6, 400 > 0</strong>. Adding <strong>6, 400</strong> to both sides of this inequality yields <strong>b² > 6, 400</strong>. Taking the square root of both sides of <strong>b² > 6, 400</strong> yields two possible inequalities: <strong>b < −80</strong> or <strong>b > 80</strong>. Of the choices, only choice A satisfies <strong>b < −80</strong> or <strong>b > 80</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "66bce0c1",
    type: "mcq",
    questionHtml: "What is the solution set of the equation above?",
    choices: [
      { label: "A", text: "<strong>set consisting of − 1</strong>" },
      { label: "B", text: "<strong>set consisting of 5</strong>" },
      { label: "C", text: "<strong>set consisting of − 1 and 5</strong>" },
      { label: "D", text: "<strong>set consisting of 0 −1, and 5</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Subtracting 4 from both sides of <strong>the √ 2 x + 6, end root + 4 = x + 3</strong> isolates the radical expression on the left side of the equation as follows: <strong>the √ 2 x + 6, end root = x − 1</strong>. Squaring both sides of <strong>the √ 2 x + 6, end root = x − 1</strong> yields <strong>2 x + 6 = x² − 2 x + 1</strong>. This equation can be rewritten as a quadratic equation in standard form: <strong>x² − 4 x − 5 = 0</strong>. One way to solve this quadratic equation is to factor the expression <strong>x² − 4 x − 5</strong> by identifying two numbers with a sum of <strong>−4</strong> and a product of <strong>−5</strong>. These numbers are <strong>−5</strong> and 1. So the quadratic equation can be factored as <strong>(x − 5, ) · (x + 1, ) = 0</strong>. It follows that 5 and <strong>−1</strong> are the solutions to the quadratic equation. However, the solutions must be verified by checking whether 5 and<strong>−1</strong> satisfy the original equation, <strong>the √ 2 x + 6, end root + 4 = x + 3</strong>. When <strong>x = −1</strong>, the original equation gives <strong>the √ 2 · −1 + 6, end root + 4 = −1 + 3</strong>, or <strong>6 = 2</strong>, which is false. Therefore, <strong>−1</strong> does not satisfy the original equation. When <strong>x = 5</strong>, the original equation gives <strong>the √ 2 · 5 + 6, end root + 4 = 5 + 3</strong>, or <strong>8 = 8</strong>, which is true. Therefore, <strong>x = 5</strong> is the only solution to the original equation, and so the solution set is <strong>5</strong>.Choices A, C, and D are incorrect because each of these sets contains at least one value that results in a false statement when substituted into the given equation. For instance, in choice D, when 0 is substituted for x into the given equation, the result is <strong>the √ 2 · 0 + 6 + 4, end root = 0 + 3</strong>, or <strong>the √ 6, end root + 4 = 3</strong>. This is not a true statement, so 0 is not a solution to the given equation.",
    hasFigure: false,
  },
  {
    id: "6ce95fc8",
    type: "mcq",
    questionHtml: "Which of the following is a solution to the equation above?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>1 − the √ 11</strong>" },
      { label: "C", text: "<strong>one half + the √ 11</strong>" },
      {
        label: "D",
        text: "<strong>the fraction with numerator 1 + the √ 11, and denominator 2</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. A quadratic equation in the form <strong>a, x² + b x + c = 0</strong>, where a, b, and c are constants, can be solved using the quadratic formula: <strong>x = the fraction with numerator − b + or − the √ b² − 4 a c, end root, and denominator 2 a, end fraction</strong>. Subtracting <strong>2 x + 3</strong> from both sides of the given equation yields <strong>2 x² − 2 x − 5 = 0</strong>. Applying the quadratic formula, where <strong>a = 2</strong>, <strong>b = −2</strong>, and <strong>c = −5</strong>, yields <strong>x = the fraction with numerator negative, (−2, ) + or − the √, (−2, ), ² − 4 · 2 · −5, end root, and denominator 2 · 2, end fraction</strong>. This can be rewritten as <strong>x = the fraction with numerator 2 + or − the √ 44, end root, and denominator 4</strong>. Since <strong>the √ 44 = the √ 2² · 11, end root</strong>, or <strong>2 · the √ 11</strong>, the equation can be rewritten as <strong>x = the fraction with numerator 2 + or − 2 · the √ 11, end root, and denominator 4</strong>. Dividing 2 from both the numerator and denominator yields <strong>the fraction with numerator 1 + the √ 11, end root, and denominator 2</strong> or <strong>the fraction with numerator 1 − the √ 11, end root, and denominator 2</strong>. Of these two solutions, only <strong>the fraction with numerator 1 + the √ 11, end root, and denominator 2</strong> is present among the choices. Thus, the correct choice is D.Choice A is incorrect and may result from a computational or conceptual error. Choice B is incorrect and may result from using <strong>x = the fraction with numerator − b + or − the √ b² − 4 a c, end root, and denominator a, end fraction</strong> instead of <strong>x = the fraction with numerator − b + or − the √ b² − 4 a c, end root, and denominator 2 a, end fraction</strong> as the quadratic formula. Choice C is incorrect and may result from rewriting <strong>the √ 44</strong> as <strong>4 · the √ 11</strong> instead of <strong>2 · the √ 11</strong>.",
    hasFigure: false,
  },
  {
    id: "71014fb1",
    type: "mcq",
    questionHtml:
      "<strong>(x − 1)² = −4</strong><br>How many distinct real solutions does the given equation have?",
    choices: [
      { label: "A", text: "Exactly one" },
      { label: "B", text: "Exactly two" },
      { label: "C", text: "Infinitely many" },
      { label: "D", text: "Zero" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Any quantity that is positive or negative in value has a positive value when squared. Therefore, the left-hand side of the given equation is either positive or zero for any value of <em>(expression)</em>. Since the right-hand side of the given equation is negative, there is no value of <em>(expression)</em> for which the given equation is true. Thus, the number of distinct real solutions for the given equation is zero.<br>Choices A, B, and C are incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "77c0cced",
    type: "mcq",
    questionHtml:
      "<strong>y = 2 x² − 21 x + 64</strong><br><strong>y = 3 x + a</strong><br>In the given system of equations, <strong>a</strong> is a constant. The graphs of the equations in the given system intersect at exactly one point, <strong>(x, y)</strong>, in the xy-plane. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>−8</strong>" },
      { label: "B", text: "<strong>−6</strong>" },
      { label: "C", text: "<strong>6</strong>" },
      { label: "D", text: "<strong>8</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that the graphs of the equations in the given system intersect at exactly one point, <strong>(x, y)</strong>, in the xy-plane. Therefore, <strong>(x, y)</strong> is the only solution to the given system of equations. The given system of equations can be solved by subtracting the second equation, <strong>y = 3 x + a</strong>, from the first equation, <strong>y = 2 x² − 21 x + 64</strong>. This yields <strong>y − y = (2 x² − 21 x + 64) − (3 x + a)</strong>, or <strong>0 = 2 x² − 24 x + 64 − a</strong>. Since the given system has only one solution, this equation has only one solution. A quadratic equation in the form <strong>r x² + s x + t = 0</strong>, where <strong>r</strong>, <strong>s</strong>, and <strong>t</strong> are constants, has one solution if and only if the discriminant, <strong>s² − 4 r t</strong>, is equal to zero. Substituting <strong>2</strong> for <strong>r</strong>, <strong>−24</strong> for <strong>s</strong>, and <strong>−a + 64</strong> for <strong>t</strong> in the expression <strong>s² − 4 r t</strong> yields <strong>(−24)² − (4) (2) (64 − a)</strong>. Setting this expression equal to zero yields <strong>(−24)² − (4) (2) (64 − a) = 0</strong>, or <strong>8 a + 64 = 0</strong>. Subtracting <strong>64</strong> from both sides of this equation yields <strong>8 a = −64</strong>. Dividing both sides of this equation by <strong>8</strong> yields <strong>a = −8</strong>. Substituting <strong>−8</strong> for <strong>a</strong> in the equation <strong>0 = 2 x² − 24 x + 64 − a</strong> yields <strong>0 = 2 x² − 24 x + 64 + 8</strong>, or <strong>0 = 2 x² − 24 x + 72</strong>. Factoring <strong>2</strong> from the right-hand side of this equation yields <strong>0 = 2 (x² − 12 x + 36)</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>0 = x² − 12 x + 36</strong>, which is equivalent to <strong>0 = (x − 6) (x − 6)</strong>, or <strong>0 = (x − 6)²</strong>. Taking the square root of both sides of this equation yields <strong>0 = x − 6</strong>. Adding <strong>6</strong> to both sides of this equation yields <strong>x = 6</strong>. <br>Choice A is incorrect. This is the value of <strong>a</strong>, not <strong>x</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "7bd10ef3",
    type: "mcq",
    questionHtml:
      "In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?",
    choices: [
      { label: "A", text: "<strong>−3</strong>" },
      { label: "B", text: "<strong>−1</strong>" },
      { label: "C", text: "<strong>1</strong>" },
      { label: "D", text: "<strong>3</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The number of solutions to any quadratic equation in the form <strong>a, x² + b x + c = 0</strong>, where a, b, and c are constants, can be found by evaluating the expression <strong>b² − 4 a, c</strong>, which is called the discriminant. If the value of <strong>b² − 4 a, c</strong> is a positive number, then there will be exactly two real solutions to the equation. If the value of <strong>b² − 4 a, c</strong> is zero, then there will be exactly one real solution to the equation. Finally, if the value of <strong>b² − 4 a, c</strong> is negative, then there will be no real solutions to the equation.The given equation <strong>2 x² − 4 x = t</strong> is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives <strong>2 x² − 4 x − t = 0</strong>. In this form, <strong>a = 2</strong>, <strong>b = −4</strong>, and <strong>c = −t</strong>. The values of t for which the equation has no real solutions are the same values of t for which the discriminant of this equation is a negative value. The discriminant is equal to <strong>(−4, ), ² − 4 · 2 · −t</strong>; therefore, <strong>(−4, ), ² − 4 · 2 · −t < 0</strong>. Simplifying the left side of the inequality gives <strong>16 + 8, t < 0</strong>. Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives <strong>t < −2</strong>. Of the values given in the options, <strong>−3</strong> is the only value that is less than <strong>−2</strong>. Therefore, choice A must be the correct answer.<br>Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.",
    hasFigure: false,
  },
  {
    id: "7dbd46d9",
    type: "mcq",
    questionHtml:
      "<strong>8 x + y = −11</strong><br><strong>2 x² = y + 341</strong><br>The graphs of the equations in the given system of equations intersect at the point <strong>(x, y)</strong> in the <strong>xy</strong>-plane. What is a possible value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>−15</strong>" },
      { label: "B", text: "<strong>−11</strong>" },
      { label: "C", text: "<strong>2</strong>" },
      { label: "D", text: "<strong>8</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the graphs of the equations in the given system of equations intersect at the point <strong>(x, y)</strong>. Therefore, this intersection point is a solution to the given system. The solution can be found by isolating <strong>y</strong> in each equation. The given equation <strong>8 x + y = −11</strong> can be rewritten to isolate <strong>y</strong> by subtracting <strong>8 x</strong> from both sides of the equation, which gives <strong>y = − 8 x − 11</strong>. The given equation <strong>2 x² = y + 341</strong> can be rewritten to isolate <strong>y</strong> by subtracting <strong>341</strong> from both sides of the equation, which gives <strong>2 x² − 341 = y</strong>. With each equation solved for <strong>y</strong>, the value of <strong>y</strong> from one equation can be substituted into the other, which gives <strong>2 x² − 341 = − 8 x − 11</strong>. Adding <strong>8 x</strong> and <strong>11</strong> to both sides of this equation results in <strong>2 x² + 8 x − 330 = 0</strong>. Dividing both sides of this equation by <strong>2</strong> results in <strong>x² + 4 x − 165 = 0</strong>. This equation can be rewritten by factoring the left-hand side, which yields <strong>(x + 15) (x − 11) = 0</strong>. By the zero-product property, if <strong>(x + 15) (x − 11) = 0</strong>, then <strong>(x + 15) = 0</strong>, or <strong>(x − 11) = 0</strong>. It follows that <strong>x = −15</strong>, or <strong>x = 11</strong>. Since only <strong>−15</strong> is given as a choice, a possible value of <strong>x</strong> is <strong>−15</strong>. <br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "97e50fa2",
    type: "spr",
    questionHtml:
      "The graph of the function f, defined by <strong>f of x = −one-half · (x − 4, ), ² + 10</strong>, is shown in the xy-plane above. If the function g (not shown) is defined by <strong>g of x = −x + 10</strong>, what is one possible value of a such that <strong>f of a = g of a</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is either 2 or 8. Substituting <strong>x = a</strong> in the definitions for f and g gives <strong>f of a = −one half · (a − 4, ), ² + 10</strong> and <strong>g of a = −a + 10</strong>, respectively. If <strong>f of a = g of a</strong>, then <strong>−one half · (a − 4, ), ² + 10 = −a + 10</strong>. Subtracting 10 from both sides of this equation gives <strong>−one half · (a − 4, ), ² = −a</strong>. Multiplying both sides by <strong>−2</strong> gives <strong>(a − 4, ), ² = 2 a</strong>. Expanding <strong>(a − 4, ), ²</strong> gives <strong>a, ² − 8 a + 16 = 2 a</strong>. Combining the like terms on one side of the equation gives <strong>a, ² − 10 a + 16 = 0</strong>. One way to solve this equation is to factor <strong>a, ² − 10 a + 16</strong> by identifying two numbers with a sum of <strong>−10</strong> and a product of 16. These numbers are <strong>−2</strong> and <strong>−8</strong>, so the quadratic equation can be factored as <strong>(a − 2, ) · (a − 8, ) = 0</strong>. Therefore, the possible values of a are either 2 or 8. Note that 2 and 8 are examples of ways to enter a correct answer.Alternate approach: Graphically, the condition <strong>f of a = g of a</strong> implies the graphs of the functions <strong>y = f of x</strong> and <strong>y = g of x</strong> intersect at <strong>x = a</strong>. The graph <strong>y = f of x</strong> is given, and the graph of <strong>y = g of x</strong> may be sketched as a line with y-intercept 10 and a slope of <strong>−1</strong> (taking care to note the different scales on each axis). These two graphs intersect at <strong>x = 2</strong> and <strong>x = 8</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/97e50fa2_img1.png",
  },
  {
    id: "9cb9beec",
    type: "spr",
    questionHtml:
      "<strong>y = −1.50</strong><br><strong>y = x² + 8 x + a</strong><br>In the given system of equations, <strong>a</strong> is a positive constant. The system has exactly one distinct real solution. What is the value of <strong>a</strong>?",
    choices: [],
    correctAnswer: "14.5, 29/2",
    explanation:
      "The correct answer is <strong>(29) / (2)</strong>. According to the first equation in the given system, the value of <strong>y</strong> is <strong>−1.5</strong>. Substituting <strong>−1.5</strong> for <strong>y</strong> in the second equation in the given system yields <strong>−1.5 = x² + 8 x + a</strong>. Adding <strong>1.5</strong> to both sides of this equation yields <strong>0 = x² + 8 x + a + 1.5</strong>. If the given system has exactly one distinct real solution, it follows that <strong>0 = x² + 8 x + a + 1.5</strong> has exactly one distinct real solution. A quadratic equation in the form <strong>0 = p x² + q x + r</strong>, where <strong>p</strong>, <strong>q</strong>, and <strong>r</strong> are constants, has exactly one distinct real solution if and only if the discriminant, <strong>q² − 4 p r</strong>, is equal to <strong>0</strong>. The equation <strong>0 = x² + 8 x + a + 1.5</strong> is in this form, where <strong>p = 1</strong>, <strong>q = 8</strong>, and <strong>r = a + 1.5</strong>. Therefore, the discriminant of the equation <strong>0 = x² + 8 x + a + 1.5</strong> is <strong>(8)² − 4 (1) (a + 1.5)</strong>, or <strong>58 − 4 a</strong>. Setting the discriminant equal to <strong>0</strong> to solve for <strong>a</strong> yields <strong>58 − 4 a = 0</strong>. Adding <strong>4 a</strong> to both sides of this equation yields <strong>58 = 4 a</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>(58) / (4) = a</strong>, or <strong>(29) / (2) = a</strong>. Therefore, if the given system of equations has exactly one distinct real solution, the value of <strong>a</strong> is <strong>(29) / (2)</strong>. Note that 29/2 and 14.5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "a54753ca",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the graph of the equation <strong>y = − x² + 9 x − 100</strong> intersects the line <strong>y = c</strong> at exactly one point. What is the value of <strong>c</strong>?",
    choices: [
      { label: "A", text: "<strong>−(481) / (4)</strong>" },
      { label: "B", text: "<strong>−100</strong>" },
      { label: "C", text: "<strong>−(319) / (4)</strong>" },
      { label: "D", text: "<strong>−nine halves</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. In the xy-plane, the graph of the line <strong>y = c</strong> is a horizontal line that crosses the y-axis at <strong>y = c</strong> and the graph of the quadratic equation <strong>y = − x² + 9 x − 100</strong> is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of <strong>c</strong> should be equal to the y-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, <strong>y = a (x − h)² + k</strong>, the vertex of its graph in the xy-plane is <strong>(h, k)</strong>. The given quadratic equation, <strong>y = − x² + 9 x − 100</strong>, can be rewritten as <strong>y = − (x² − 2 (nine halves) x + (nine halves)²) + (nine halves)² − 100</strong>, or <strong>y = − (x − nine halves)² + (−(319) / (4))</strong>. Thus, the value of <strong>c</strong> is equal to <strong>−(319) / (4)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b03adde3",
    type: "mcq",
    questionHtml:
      "If <strong>u − 3 = the fraction 6 over, t − 2, end fraction</strong>, what is t in terms of u ?",
    choices: [
      { label: "A", text: "<strong>t = the fraction 1 over u</strong>" },
      { label: "B", text: "<strong>t = the fraction 2 u + 9, over u</strong>" },
      {
        label: "C",
        text: "<strong>t = the fraction 1 over, u − 3, end fraction</strong>",
      },
      {
        label: "D",
        text: "<strong>t = the fraction 2 u, over u − 3, end fraction</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Multiplying both sides of the given equation by <strong>t − 2</strong> yields <strong>(t − 2, ) · (u − 3, ) = 6</strong>. Dividing both sides of this equation by <strong>u − 3</strong> yields <strong>t − 2 = the fraction with numerator 6, and denominator u − 3, end fraction</strong>. Adding 2 to both sides of this equation yields <strong>t = the fraction with numerator 6, and denominator u − 3, end fraction + 2</strong>, which can be rewritten as <strong>t = the fraction with numerator 6, and denominator u − 3, end fraction + the fraction with numerator 2 · (u − 3, ), and denominator u − 3, end fraction</strong>. Since the fractions on the right-hand side of this equation have a common denominator, adding the fractions yields <strong>t = the fraction with numerator 6 + 2 · (u − 3, ), and denominator u − 3, end fraction</strong>. Applying the distributive property to the numerator on the right-hand side of this equation yields <strong>t = the fraction with numerator 6 + 2 u − 6, and denominator u − 3, end fraction</strong>, which is equivalent to <strong>t = the fraction with numerator 2 u, and denominator u − 3, end fraction</strong>.Choices A, B, and C are incorrect and may result from various misconceptions or miscalculations.",
    hasFigure: false,
  },
  {
    id: "ba0edc30",
    type: "mcq",
    questionHtml:
      "<strong>x² − 2 x − 9 = 0</strong><br>One solution to the given equation can be written as <strong>1 + √(k)</strong>, where <strong>k</strong> is a constant. What is the value of <strong>k</strong>?",
    choices: [
      { label: "A", text: "<strong>8</strong>" },
      { label: "B", text: "<strong>10</strong>" },
      { label: "C", text: "<strong>20</strong>" },
      { label: "D", text: "<strong>40</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Adding <strong>9</strong> to each side of the given equation yields <strong>x² − 2 x = 9</strong>. To complete the square, adding <strong>1</strong> to each side of this equation yields <strong>x² − 2 x + 1 = 9 + 1</strong>, or <strong>(x − 1)² = 10</strong>. Taking the square root of each side of this equation yields <strong>x − 1 = + or − √(10)</strong>. Adding <strong>1</strong> to each side of this equation yields <strong>x = 1 + or − √(10)</strong>. Since it's given that one of the solutions to the equation can be written as <strong>1 + √(k)</strong>, the value of <strong>k</strong> must be <strong>10</strong>.<br>Alternate approach: It's given that <strong>1 + √(k)</strong> is a solution to the given equation. It follows that <strong>x = 1 + √(k)</strong>. Substituting <strong>1 + √(k)</strong> for <strong>x</strong> in the given equation yields <strong>(1 + √(k))² − 2 (1 + √(k)) − 9 = 0</strong>, or <strong>(1 + √(k)) (1 + √(k)) − 2 (1 + √(k)) − 9 = 0</strong>. Expanding the products on the left-hand side of this equation yields <strong>1 + 2 √(k) + k − 2 − 2 √(k) − 9 = 0</strong>, or <strong>k − 10 = 0</strong>. Adding <strong>10</strong> to each side of this equation yields <strong>k = 10</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "c303ad23",
    type: "spr",
    questionHtml:
      "If <strong>3 x² − 18 x − 15 = 0</strong>, what is the value of <strong>x² − 6 x</strong>?",
    choices: [],
    correctAnswer: "5",
    explanation:
      "The correct answer is <strong>5</strong>. Dividing each side of the given equation by <strong>3</strong> yields <strong>x² − 6 x − 5 = 0</strong>. Adding <strong>5</strong> to each side of this equation yields <strong>x² − 6 x = 5</strong>. Therefore, if <strong>3 x² − 18 x − 15 = 0</strong>, the value of <strong>x² − 6 x</strong> is <strong>5</strong>.",
    hasFigure: false,
  },
  {
    id: "d0a53ef5",
    type: "spr",
    questionHtml:
      "<strong>√((x − 2)²) = √(3 x + 34)</strong><br>What is the smallest solution to the given equation?",
    choices: [],
    correctAnswer: "-3",
    explanation:
      "The correct answer is <strong>−3</strong>. Squaring both sides of the given equation yields <strong>(x − 2)² = 3 x + 34</strong>, which can be rewritten as <strong>x² − 4 x + 4 = 3 x + 34</strong>. Subtracting <strong>3 x</strong> and <strong>34</strong> from both sides of this equation yields <strong>x² − 7 x − 30 = 0</strong>. This quadratic equation can be rewritten as <strong>(x − 10) (x + 3) = 0</strong>. According to the zero product property, <strong>(x − 10) (x + 3)</strong> equals zero when either <strong>x − 10 = 0</strong> or <strong>x + 3 = 0</strong>. Solving each of these equations for <strong>x</strong> yields <strong>x = 10</strong> or <strong>x = −3</strong>. Therefore, the given equation has two solutions, <strong>10</strong> and <strong>−3</strong>. Of these two solutions, <strong>−3</strong> is the smallest solution to the given equation.",
    hasFigure: false,
  },
  {
    id: "e9349667",
    type: "mcq",
    questionHtml:
      "If <strong>the ordered pair x subscript 1, y subscript 1</strong> and <strong>the ordered pair x subscript 2, y subscript 2</strong> are the two solutions to the system of equations above, what is the value of <strong>y subscript 1 + y subscript 2</strong> ?",
    choices: [
      { label: "A", text: "<strong>−3</strong>" },
      { label: "B", text: "<strong>−2</strong>" },
      { label: "C", text: "<strong>−1</strong>" },
      { label: "D", text: "<strong>1</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for y gives y = –x – 1. Substituting the expression –x – 1 for y into the first equation gives –x – 1 = x2 + 2x + 1. Adding x + 1 to both sides of the equation yields x2 + 3x + 2 = 0. The left-hand side of the equation can be factored by finding two numbers whose sum is 3 and whose product is 2, which gives (x + 2)(x + 1) = 0. Setting each factor equal to 0 yields x + 2 = 0 and x + 1 = 0, and solving for x yields x = –2 or x = –1. These values of x can be substituted for x in the equation y = –x – 1 to find the corresponding y-values: y = –(–2) – 1 = 2 – 1 = 1 and y = –(–1) – 1 = 1 – 1 = 0. It follows that (–2, 1) and (–1, 0) are the solutions to the given system of equations. Therefore, (x1, y1) = (–2, 1), (x2, y2) = (–1, 0), and y1 + y2 = 1 + 0 = 1.<br> <br> <br>Choice A is incorrect. The solutions to the system of equations are (x1, y1) = (–2, 1) and (x2, y2) = (–1, 0). Therefore, –3 is the sum of the x-coordinates of the solutions, not the sum of the y-coordinates of the solutions. Choices B and C are incorrect and may be the result of computation or substitution errors.",
    hasFigure: false,
  },
  {
    id: "ebb717ab",
    type: "spr",
    questionHtml:
      "<strong>x² − 34 x + c = 0</strong><br>In the given equation, <strong>c</strong> is a constant. The equation has no real solutions if <strong>c > n</strong>. What is the least possible value of <strong>n</strong>?",
    choices: [],
    correctAnswer: "289",
    explanation:
      "The correct answer is <strong>289</strong>. A quadratic equation of the form <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, has no real solutions when the value of the discriminant, <strong>b² − 4 a c</strong>, is less than <strong>0</strong>. In the given equation, <strong>x² − 34 x + c = 0</strong>, <strong>a = 1</strong> and <strong>b = −34</strong>. Therefore, the discriminant of the given equation can be expressed as <strong>(−34)² − 4 (1) (c)</strong>, or <strong>1, 156 − 4 c</strong>. It follows that the given equation has no real solutions when <strong>1, 156 − 4 c < 0</strong>. Adding <strong>4 c</strong> to both sides of this inequality yields <strong>1, 156 < 4 c</strong>. Dividing both sides of this inequality by <strong>4</strong> yields <strong>289 < c</strong>, or <strong>c > 289</strong>. It's given that the equation <strong>x² − 34 x + c = 0</strong> has no real solutions when <strong>c > n</strong>. Therefore, the least possible value of <strong>n</strong> is <strong>289</strong>.",
    hasFigure: false,
  },
  {
    id: "f2f3fa00",
    type: "spr",
    questionHtml:
      "During a 5-second time interval, the average acceleration a, in meters per second squared, of an object with an initial velocity of 12 meters per second is defined by the equation <strong>a = the fraction with numerator v subscript f, end subscript − 12, and denominator 5</strong>, where vf is the final velocity of the object in meters per second. If the equation is rewritten in the form vf = xa + y, where x and y are constants, what is the value of x ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 5. The given equation can be rewritten in the form <strong>v subscript f = x a + y</strong>, like so:<br> <strong>a = the fraction with numerator v subscript f, end subscript − 12, and denominator 5</strong><br><br> <strong>v subscript f, end subscript − 12 = 5 a</strong><br><br> <strong>v subscript f = 5 a + 12</strong><br>It follows that the value of x is 5 and the value of y is 12.",
    hasFigure: false,
  },
  {
    id: "f5aa5040",
    type: "spr",
    questionHtml:
      "In the xy-plane, a line with equation <strong>2 y = c</strong> for some constant <strong>c</strong> intersects a parabola at exactly one point. If the parabola has equation <strong>y = − 2 x² + 9 x</strong>, what is the value of <strong>c</strong>?",
    choices: [],
    correctAnswer: "20.25, 81/4",
    explanation:
      "The correct answer is <strong>(81) / (4)</strong>. The given linear equation is <strong>2 y = c</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>y = (c) / (2)</strong>. Substituting <strong>(c) / (2)</strong> for <strong>y</strong> in the equation of the parabola yields <strong>(c) / (2) = − 2 x² + 9 x</strong>. Adding <strong>2 x²</strong> and <strong>− 9 x</strong> to both sides of this equation yields <strong>2 x² − 9 x + (c) / (2) = 0</strong>. Since it’s given that the line and the parabola intersect at exactly one point, the equation <strong>2 x² − 9 x + (c) / (2) = 0</strong> must have exactly one solution. An equation of the form <strong>A x² + B x + C = 0</strong>, where <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> are constants, has exactly one solution when the discriminant, <strong>B² − 4 A C</strong>, is equal to <strong>0</strong>. In the equation <strong>2 x² − 9 x + (c) / (2) = 0</strong>, where <strong>A = 2</strong>, <strong>B = −9</strong>, and <strong>C = (c) / (2)</strong>, the discriminant is <strong>(−9)² − 4 (2) ((c) / (2))</strong>. Setting the discriminant equal to <strong>0</strong> yields <strong>(−9)² − 4 (2) ((c) / (2)) = 0</strong>, or <strong>81 − 4 c = 0</strong>. Adding <strong>4 c</strong> to both sides of this equation yields <strong>81 = 4 c</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>c = (81) / (4)</strong>. Note that 81/4 and 20.25 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "f65288e8",
    type: "mcq",
    questionHtml:
      "If x is a solution to the given equation, which of the following is a possible value of <strong>x + 5</strong> ?",
    choices: [
      { label: "A", text: "<strong>one half</strong>" },
      { label: "B", text: "<strong>five halves</strong>" },
      { label: "C", text: "<strong>nine halves</strong>" },
      { label: "D", text: "<strong>eleven halves</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The given equation can be rewritten as <strong>the fraction 1 over, (x + 5, ), ², end fraction = 4</strong>. Multiplying both sides of this equation by <strong>(x + 5, ), ²</strong> yields <strong>1 = 4 · (x + 5, ), ²</strong>. Dividing both sides of this equation by 4 yields <strong>one fourth = (x + 5, ), ²</strong>. Taking the square root of both sides of this equation yields <strong>one half = x + 5</strong> or <strong>−one half = x + 5</strong>. Therefore, a possible value of <strong>x + 5</strong> is <strong>one half</strong>.<br><br>Choices B, C, and D are incorrect and may result from computational or conceptual errors.",
    hasFigure: false,
  },
  {
    id: "fbb96bb1",
    type: "mcq",
    questionHtml:
      "<strong>x − 29 = (x − a) (x − 29)</strong><br>Which of the following are solutions to the given equation, where <strong>a</strong> is a constant and <strong>a > 30</strong>?<br><br><strong>a</strong><br><strong>a + 1</strong><br><strong>29</strong>",
    choices: [
      { label: "A", text: "I and II only" },
      { label: "B", text: "I and III only" },
      { label: "C", text: "II and III only" },
      { label: "D", text: "I, II, and III" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Subtracting the expression <strong>(x − 29)</strong> from both sides of the given equation yields <strong>0 = (x − a) (x − 29) − (x − 29)</strong>, which can be rewritten as <strong>0 = (x − a) (x − 29) + (−1) (x − 29)</strong>. Since the two terms on the right-hand side of this equation have a common factor of <strong>(x − 29)</strong>, it can be rewritten as <strong>0 = (x − 29) (x − a + (−1))</strong>, or <strong>0 = (x − 29) (x − a − 1)</strong>. Since <strong>x − a − 1</strong> is equivalent to <strong>x − (a + 1)</strong>, the equation <strong>0 = (x − 29) (x − a − 1)</strong> can be rewritten as <strong>0 = (x − 29) (x − (a + 1))</strong>. By the zero product property, it follows that <strong>x − 29 = 0</strong> or <strong>x − (a + 1) = 0</strong>. Adding <strong>29</strong> to both sides of the equation <strong>x − 29 = 0</strong> yields <strong>x = 29</strong>. Adding <strong>a + 1</strong> to both sides of the equation <strong>x − (a + 1) = 0</strong> yields <strong>x = a + 1</strong>. Therefore, the two solutions to the given equation are <strong>29</strong> and <strong>a + 1</strong>. Thus, only <strong>a + 1</strong> and <strong>29</strong>, not <strong>a</strong>, are solutions to the given equation.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "fc3d783a",
    type: "spr",
    questionHtml:
      "In the <em>(expression)</em>-plane, a line with equation <em>(expression)</em> intersects a parabola at exactly one point. If the parabola has equation <em>(expression)</em>, where <em>(expression)</em> is a positive constant, what is the value of <em>(expression)</em>?",
    choices: [],
    correctAnswer: "6",
    explanation:
      "The correct answer is <strong>6</strong>. It’s given that a line with equation <strong>2 y = 4 . 5</strong> intersects a parabola with equation <strong>y = − 4 x² + b x</strong>, where <strong>b</strong> is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of <strong>2 y = 4 . 5</strong> and <strong>y = − 4 x² + b x</strong> has exactly one solution. Dividing both sides of the equation of the line by <strong>2</strong> yields <strong>y = 2 . 25</strong>. Substituting <strong>2.25</strong> for <strong>y</strong> in the equation of the parabola yields <strong>2 . 25 = − 4 x² + b x</strong>. Adding <strong>4 x²</strong> and subtracting <strong>b x</strong> from both sides of this equation yields <strong>4 x² − b x + 2 . 25 = 0</strong>. A quadratic equation in the form of <strong>a x² + b x + c = 0</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, has exactly one solution when the discriminant, <strong>b² − 4 a c</strong>, is equal to zero. Substituting <strong>4</strong> for <strong>a</strong> and <strong>2.25</strong> for <strong>c</strong> in the expression <strong>b² − 4 a c</strong> and setting this expression equal to <strong>0</strong> yields <strong>b² − 4 (4) (2 . 25) = 0</strong>, or <strong>b² − 36 = 0</strong>. Adding <strong>36</strong> to each side of this equation yields <strong>b² = 36</strong>. Taking the square root of each side of this equation yields <strong>b = + or − 6</strong>. It’s given that <strong>b</strong> is positive, so the value of <strong>b</strong> is <strong>6</strong>.",
    hasFigure: false,
  },
  {
    id: "fc3dfa26",
    type: "mcq",
    questionHtml: "What value of x satisfies the equation above?",
    choices: [
      { label: "A", text: "<strong>−3</strong>" },
      { label: "B", text: "<strong>−one half</strong>" },
      { label: "C", text: "<strong>one half</strong>" },
      { label: "D", text: "<strong>3</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Each fraction in the given equation can be expressed with the common denominator <strong>x² − 9</strong>. Multiplying <strong>the fraction with numerator 2 x, and denominator x + 3, end fraction</strong> by <strong>the fraction with numerator x − 3, and denominator x − 3, end fraction</strong> yields <strong>the fraction with numerator 2 x² − 6, and denominator x² − 9, end fraction</strong>, and multiplying <strong>the fraction with numerator 1, and denominator x − 3, end fraction</strong> by <strong>the fraction with numerator x + 3, and denominator x + 3, end fraction</strong> yields <strong>the fraction with numerator x + 3, and denominator x² − 9, end fraction</strong>. Therefore, the given equation can be written as <strong>the fraction with numerator 4 x², and denominator x² − 9 − the fraction with numerator 2 x² − 6 x, and denominator x² − 9, end fraction = the fraction with numerator x + 3, and denominator x² − 9, end fraction</strong>. Multiplying each fraction by the denominator results in the equation <strong>4 x² − (2 x² − 6 x, ) = x + 3</strong>, or <strong>2 x² + 6 x = x + 3</strong>. This equation can be solved by setting a quadratic expression equal to 0, then solving for x. Subtracting <strong>x + 3</strong> from both sides of this equation yields <strong>2 x² + 5 x − 3 = 0</strong>. The expression <strong>2 x² + 5 x − 3</strong> can be factored, resulting in the equation <strong>(2 x − 1, ) · (x + 3, ) = 0</strong>. By the zero product property, <strong>2 x − 1 = 0</strong> or <strong>x + 3 = 0</strong>. To solve for x in <strong>2 x − 1 = 0</strong>, 1 can be added to both sides of the equation, resulting in <strong>2 x = 1</strong>. Dividing both sides of this equation by 2 results in <strong>x = one half</strong>. Solving for x in <strong>x + 3 = 0</strong> yields <strong>x = −3</strong>. However, this value of x would result in the second fraction of the original equation having a denominator of 0. Therefore, <strong>x = −3</strong> is an extraneous solution. Thus, the only value of x that satisfies the given equation is <strong>x = one half</strong>.Choice A is incorrect and may result from solving <strong>x + 3 = 0</strong> but not realizing that this solution is extraneous because it would result in a denominator of 0 in the second fraction. Choice B is incorrect and may result from a sign error when solving <strong>2 x − 1 = 0</strong> for x. Choice D is incorrect and may result from a calculation error.",
    hasFigure: false,
  },
  {
    id: "ff2e5c76",
    type: "mcq",
    questionHtml:
      "<strong>x² − 40 x − 10 = 0</strong><br>What is the sum of the solutions to the given equation?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>10</strong>" },
      { label: "D", text: "<strong>40</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Adding <strong>10</strong> to each side of the given equation yields <strong>x² − 40 x = 10</strong>. To complete the square, adding <strong>((40) / (2))²</strong>, or <strong>20²</strong>, to each side of this equation yields <strong>x² − 40 x + 20² = 10 + 20²</strong>, or <strong>(x − 20)² = 410</strong>. Taking the square root of each side of this equation yields <strong>x − 20 = + or − √(410)</strong>. Adding <strong>20</strong> to each side of this equation yields <strong>x = 20 + or − √(410)</strong>. Therefore, the solutions to the given equation are <strong>x = 20 + √(410)</strong> and <strong>x = 20 − √(410)</strong>. The sum of these solutions is <strong>(20 + √(410)) + (20 − √(410))</strong>, or <strong>40</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];
