feat(lessons): add lessons from client db

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import React, { useRef, useState, useEffect } from "react";
import {
ArrowDown,
Check,
BookOpen,
TrendingUp,
Grid,
RefreshCw,
} from "lucide-react";
import ParabolaWidget from "../../../components/lessons/ParabolaWidget";
import DiscriminantWidget from "../../../components/lessons/DiscriminantWidget";
import LinearQuadraticSystemWidget from "../../../components/lessons/LinearQuadraticSystemWidget";
import Quiz from "../../../components/lessons/Quiz";
import { QUADRATIC_EQ_QUIZ_DATA } from "../../../utils/constants";
import { Frac } from "../../../components/Math";
interface LessonProps {
onFinish?: () => void;
}
const QuadraticEquationsLesson: React.FC<LessonProps> = ({ onFinish }) => {
const [activeSection, setActiveSection] = useState(0);
const sectionsRef = useRef<(HTMLElement | null)[]>([]);
const scrollToSection = (index: number) => {
setActiveSection(index);
sectionsRef.current[index]?.scrollIntoView({
behavior: "smooth",
block: "start",
});
};
useEffect(() => {
const observer = new IntersectionObserver(
(entries) => {
entries.forEach((entry) => {
if (entry.isIntersecting) {
const index = sectionsRef.current.indexOf(
entry.target as HTMLElement,
);
if (index !== -1) setActiveSection(index);
}
});
},
{ rootMargin: "-20% 0px -60% 0px" },
);
sectionsRef.current.forEach((section) => {
if (section) observer.observe(section);
});
return () => observer.disconnect();
}, []);
const SectionMarker = ({
index,
title,
icon: Icon,
}: {
index: number;
title: string;
icon: any;
}) => {
const isActive = activeSection === index;
const isPast = activeSection > index;
return (
<button
onClick={() => scrollToSection(index)}
className={`flex items-center gap-3 p-3 w-full rounded-lg transition-all ${isActive ? "bg-white shadow-md border border-violet-100" : "hover:bg-slate-100"}`}
>
<div
className={`w-8 h-8 rounded-full flex items-center justify-center shrink-0 ${isActive ? "bg-violet-600 text-white" : isPast ? "bg-violet-400 text-white" : "bg-slate-200 text-slate-500"}`}
>
{isPast ? (
<Check className="w-4 h-4" />
) : (
<Icon className="w-4 h-4" />
)}
</div>
<div className="text-left">
<p
className={`text-sm font-bold ${isActive ? "text-violet-900" : "text-slate-600"}`}
>
{title}
</p>
</div>
</button>
);
};
return (
<div className="flex flex-col lg:flex-row min-h-screen">
<aside className="w-full lg:w-64 lg:fixed lg:top-20 lg:bottom-0 lg:overflow-y-auto p-4 border-r border-slate-200 bg-slate-50 z-0 hidden lg:block">
<nav className="space-y-2">
<SectionMarker
index={0}
title="Parabolas & Forms"
icon={TrendingUp}
/>
<SectionMarker
index={1}
title="Solving & Discriminant"
icon={RefreshCw}
/>
<SectionMarker index={2} title="Systems" icon={Grid} />
<SectionMarker index={3} title="Practice" icon={BookOpen} />
</nav>
</aside>
<div className="flex-1 lg:ml-64 p-6 md:p-12 max-w-4xl mx-auto">
{/* Section 1: Parabolas & Forms */}
<section
ref={(el) => {
sectionsRef.current[0] = el;
}}
className="min-h-screen flex flex-col justify-center mb-24 pt-20 lg:pt-0"
>
<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
Parabolas & Quadratic Forms
</h2>
<div className="prose prose-slate text-lg text-slate-600 mb-8">
<p>
A quadratic function creates a U-shaped curve called a{" "}
<strong>parabola</strong>. The SAT uses three different but
equivalent forms each form highlights different features.
Knowing all three lets you pick the most efficient approach for
each question.
</p>
</div>
{/* Three Forms Card */}
<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-8 space-y-5">
<h3 className="text-lg font-bold text-violet-900">
The Three Quadratic Forms
</h3>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<div className="flex items-baseline gap-3 mb-2">
<span className="text-xs font-bold uppercase tracking-wider text-violet-500 bg-violet-100 px-2 py-0.5 rounded">
Standard Form
</span>
</div>
<p className="font-mono text-violet-800 font-bold text-xl text-center mb-3">
y = ax² + bx + c
</p>
<ul className="text-slate-600 text-sm space-y-1 list-disc list-inside">
<li>
<strong>a &gt; 0</strong>: parabola opens upward (U-shape);{" "}
<strong>a &lt; 0</strong>: opens downward (-shape)
</li>
<li>
<strong>|a| &gt; 1</strong>: narrow parabola;{" "}
<strong>|a| &lt; 1</strong>: wide parabola
</li>
<li>
<strong>c</strong> is the y-intercept (the value of y when x =
0)
</li>
<li>
Vertex x-coordinate: x = <Frac n="b" d="2a" />
</li>
<li>
Axis of symmetry: x = <Frac n="b" d="2a" />
</li>
</ul>
<div className="mt-3 bg-violet-50 rounded-lg p-3 text-sm">
<p className="font-semibold text-violet-800">Example:</p>
<p className="text-slate-600">
y = 2x² 8x + 6 axis of symmetry: x ={" "}
<Frac n="(8)" d="2 × 2" /> = <Frac n="8" d="4" /> ={" "}
<strong>2</strong>
</p>
</div>
</div>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<div className="flex items-baseline gap-3 mb-2">
<span className="text-xs font-bold uppercase tracking-wider text-violet-500 bg-violet-100 px-2 py-0.5 rounded">
Vertex Form
</span>
</div>
<p className="font-mono text-violet-800 font-bold text-xl text-center mb-3">
y = a(x h)² + k
</p>
<ul className="text-slate-600 text-sm space-y-1 list-disc list-inside">
<li>Vertex is at (h, k) read directly from the equation</li>
<li>
Watch the sign: y = a(x <strong>3</strong>)² + 5 has vertex
at (<strong>3</strong>, 5), not (3, 5)
</li>
<li>
k is the minimum value (if a &gt; 0) or maximum value (if a
&lt; 0) of y
</li>
<li>
Best form to use when a question asks about the vertex,
minimum, or maximum
</li>
</ul>
<div className="mt-3 bg-violet-50 rounded-lg p-3 text-sm">
<p className="font-semibold text-violet-800">Example:</p>
<p className="text-slate-600">
y = 3(x + 2)² + 7 vertex at (2, 7); maximum value is{" "}
<strong>7</strong> (since a = 3 &lt; 0)
</p>
</div>
</div>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<div className="flex items-baseline gap-3 mb-2">
<span className="text-xs font-bold uppercase tracking-wider text-violet-500 bg-violet-100 px-2 py-0.5 rounded">
Factored Form
</span>
</div>
<p className="font-mono text-violet-800 font-bold text-xl text-center mb-3">
y = a(x r)(x r)
</p>
<ul className="text-slate-600 text-sm space-y-1 list-disc list-inside">
<li>x-intercepts (roots/zeros) are at x = r and x = r</li>
<li>Set each factor = 0 to find roots: x r = 0 x = r</li>
<li>
Axis of symmetry is the midpoint of the roots: x ={" "}
<Frac n="r₁ + r₂" d="2" />
</li>
<li>
Best form to use when a question asks about x-intercepts or
roots
</li>
</ul>
<div className="mt-3 bg-violet-50 rounded-lg p-3 text-sm">
<p className="font-semibold text-violet-800">Example:</p>
<p className="text-slate-600">
y = 2(x 1)(x 5) roots at x = 1 and x = 5; axis of
symmetry: x = <Frac n="1 + 5" d="2" /> = <strong>3</strong>
</p>
</div>
</div>
</div>
{/* Converting Between Forms */}
<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-8 space-y-4">
<h3 className="text-lg font-bold text-violet-900">
Converting Between Forms
</h3>
<div className="grid grid-cols-1 md:grid-cols-2 gap-4">
<div className="bg-white rounded-xl p-4 border border-violet-100">
<p className="font-bold text-violet-800 mb-2">
Standard Vertex (Completing the Square)
</p>
<div className="text-sm text-slate-600 space-y-1 font-mono">
<p>y = x² 6x + 5</p>
<p>= (x² 6x + 9) 9 + 5</p>
<p>= (x 3)² 4</p>
<p className="text-violet-700 font-bold">Vertex: (3, 4)</p>
</div>
</div>
<div className="bg-white rounded-xl p-4 border border-violet-100">
<p className="font-bold text-violet-800 mb-2">
Standard Factored
</p>
<div className="text-sm text-slate-600 space-y-1 font-mono">
<p>y = x² 6x + 5</p>
<p>Find two numbers: × = 5, + = 6</p>
<p>Those numbers: 1 and 5</p>
<p>= (x 1)(x 5)</p>
<p className="text-violet-700 font-bold">
Roots: x = 1 and x = 5
</p>
</div>
</div>
</div>
<div className="bg-red-50 border border-red-200 rounded-xl p-4 text-sm">
<p className="font-bold text-red-800 mb-1">
Common SAT Trap Vertex Form Sign
</p>
<p className="text-slate-700">
y = (x + 4)² 3 looks like h = 4, but it's actually y = (x
(4))² 3, so the vertex is at (<strong>4</strong>, 3).
Always rewrite (x + h) as (x (h)) to read the vertex
correctly.
</p>
</div>
</div>
<ParabolaWidget />
<button
onClick={() => scrollToSection(1)}
className="mt-12 group flex items-center text-violet-600 font-bold hover:text-violet-800 transition-colors"
>
Next: Solving & Discriminant{" "}
<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
</button>
</section>
{/* Section 2: Solving & Discriminant */}
<section
ref={(el) => {
sectionsRef.current[1] = el;
}}
className="min-h-screen flex flex-col justify-center mb-24"
>
<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
Solving Quadratics & The Discriminant
</h2>
<div className="prose prose-slate text-lg text-slate-600 mb-8">
<p>
There are four methods to solve a quadratic equation. Choosing the
right method quickly is an important SAT skill. Before diving in,
check the <strong>Discriminant</strong> — it tells you instantly
how many real solutions exist.
</p>
</div>
{/* Discriminant Card */}
<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-6 space-y-4">
<h3 className="text-lg font-bold text-violet-900">
The Discriminant
</h3>
<div className="bg-white rounded-xl p-4 text-center border border-violet-100">
<p className="text-2xl font-mono font-bold text-violet-800">
Δ = b² 4ac
</p>
<p className="text-slate-500 text-sm mt-1">
For ax² + bx + c = 0
</p>
</div>
<div className="overflow-x-auto">
<table className="w-full text-sm border-collapse">
<thead>
<tr className="bg-violet-200 text-violet-900">
<th className="p-3 rounded-tl-lg font-bold text-left">
Discriminant Value
</th>
<th className="p-3 font-bold text-left">
Number of Real Solutions
</th>
<th className="p-3 rounded-tr-lg font-bold text-left">
What it Means Graphically
</th>
</tr>
</thead>
<tbody>
<tr className="bg-white border-b border-violet-100">
<td className="p-3 font-bold text-green-700">Δ &gt; 0</td>
<td className="p-3 text-slate-600">
<strong>2</strong> distinct real solutions
</td>
<td className="p-3 text-slate-600">
Parabola crosses x-axis at two points
</td>
</tr>
<tr className="bg-violet-50 border-b border-violet-100">
<td className="p-3 font-bold text-amber-700">Δ = 0</td>
<td className="p-3 text-slate-600">
<strong>1</strong> repeated real solution
</td>
<td className="p-3 text-slate-600">
Parabola is tangent to x-axis (vertex on x-axis)
</td>
</tr>
<tr className="bg-white">
<td className="p-3 font-bold text-red-700">Δ &lt; 0</td>
<td className="p-3 text-slate-600">
<strong>0</strong> real solutions
</td>
<td className="p-3 text-slate-600">
Parabola does not touch x-axis
</td>
</tr>
</tbody>
</table>
</div>
</div>
{/* Four Solving Methods */}
<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-8 space-y-5">
<h3 className="text-lg font-bold text-violet-900">
Four Methods to Solve ax² + bx + c = 0
</h3>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<p className="font-bold text-violet-800 mb-3">
Method 1: Factoring (fastest when it works)
</p>
<p className="text-slate-600 text-sm mb-3">
Find two numbers that multiply to <strong>a × c</strong> and add
to <strong>b</strong>.
</p>
<div className="bg-violet-50 rounded-lg p-3 font-mono text-sm space-y-1">
<p className="text-slate-600">Solve: x² + 5x + 6 = 0</p>
<p className="text-slate-600">
Need two numbers: × = 6, + = 5 → <strong>2 and 3</strong>
</p>
<p className="text-slate-600">(x + 2)(x + 3) = 0</p>
<p className="text-violet-700 font-bold">x = 2 or x = 3</p>
</div>
</div>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<p className="font-bold text-violet-800 mb-3">
Method 2: Square Root Method (when b = 0 or vertex form)
</p>
<p className="text-slate-600 text-sm mb-3">
Isolate the squared term, then take the square root of both
sides. Remember ±.
</p>
<div className="bg-violet-50 rounded-lg p-3 font-mono text-sm space-y-1">
<p className="text-slate-600">Solve: 2x² 18 = 0</p>
<p className="text-slate-600">2x² = 18 → x² = 9</p>
<p className="text-slate-600">x = ±√9</p>
<p className="text-violet-700 font-bold">x = 3 or x = 3</p>
</div>
</div>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<p className="font-bold text-violet-800 mb-3">
Method 3: Quadratic Formula (always works)
</p>
<div className="bg-violet-50 rounded-lg p-3 text-center mb-3">
<p className="font-mono text-violet-800 font-bold text-lg">
x = <Frac n={<>b ± √(b² 4ac)</>} d="2a" />
</p>
</div>
<div className="bg-violet-50 rounded-lg p-3 font-mono text-sm space-y-1">
<p className="text-slate-600">
Solve: 2x² 3x 2 = 0 (a=2, b=3, c=2)
</p>
<p className="text-slate-600">
Δ = (3)² 4(2)(2) = 9 + 16 = 25
</p>
<p className="text-slate-600">
x = <Frac n="3 ± √25" d="4" /> = <Frac n="3 ± 5" d="4" />
</p>
<p className="text-violet-700 font-bold">x = 2 or x = −½</p>
</div>
</div>
<div className="bg-white rounded-xl p-5 border border-violet-100">
<p className="font-bold text-violet-800 mb-3">
Method 4: Completing the Square
</p>
<p className="text-slate-600 text-sm mb-3">
Rewrite into vertex form, then solve. Most useful when the SAT
asks for vertex form or when coefficients are simple.
</p>
<div className="bg-violet-50 rounded-lg p-3 font-mono text-sm space-y-1">
<p className="text-slate-600">Solve: x² + 6x + 5 = 0</p>
<p className="text-slate-600">x² + 6x = 5</p>
<p className="text-slate-600">
x² + 6x + 9 = 5 + 9 (add (<Frac n="6" d="2" />
)² = 9 to both sides)
</p>
<p className="text-slate-600">(x + 3)² = 4</p>
<p className="text-slate-600">x + 3 = ±2</p>
<p className="text-violet-700 font-bold">x = 1 or x = 5</p>
</div>
</div>
</div>
{/* SAT Strategy */}
<div className="bg-amber-50 border border-amber-200 rounded-2xl p-6 mb-8">
<h3 className="text-lg font-bold text-amber-900 mb-3">
SAT Strategy: Which Method to Use?
</h3>
<div className="space-y-2 text-sm text-slate-700">
<div className="flex gap-3 items-start">
<span className="font-bold text-violet-700 shrink-0">1.</span>
<p>
If the equation factors nicely (integer roots likely) →{" "}
<strong>Factor</strong> first. It's fastest.
</p>
</div>
<div className="flex gap-3 items-start">
<span className="font-bold text-violet-700 shrink-0">2.</span>
<p>
If the middle term is missing (bx = 0) or it's already in
vertex form → <strong>Square Root Method</strong>.
</p>
</div>
<div className="flex gap-3 items-start">
<span className="font-bold text-violet-700 shrink-0">3.</span>
<p>
If you can't factor quickly or need exact answers {" "}
<strong>Quadratic Formula</strong>.
</p>
</div>
<div className="flex gap-3 items-start">
<span className="font-bold text-violet-700 shrink-0">4.</span>
<p>
If the question asks to "rewrite in vertex form" or "find the
vertex" <strong>Complete the Square</strong>.
</p>
</div>
</div>
</div>
<DiscriminantWidget />
<button
onClick={() => scrollToSection(2)}
className="mt-12 group flex items-center text-violet-600 font-bold hover:text-violet-800 transition-colors"
>
Next: Linear-Quadratic Systems{" "}
<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
</button>
</section>
{/* Section 3: Systems */}
<section
ref={(el) => {
sectionsRef.current[2] = el;
}}
className="min-h-screen flex flex-col justify-center mb-24"
>
<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
Linear-Quadratic Systems
</h2>
<div className="prose prose-slate text-lg text-slate-600 mb-8">
<p>
When a line intersects a parabola, the system can have 0, 1, or 2
solutions. The key strategy is to substitute the linear equation
into the quadratic, rearrange everything to one side to form a new
quadratic, then use the discriminant to determine the number of
intersections.
</p>
</div>
<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-8 space-y-5">
<h3 className="text-lg font-bold text-violet-900">
Strategy: 4-Step Process
</h3>
<div className="space-y-3">
{[
{
step: "1",
title: "Write out the system",
body: "You have a linear equation (y = mx + b) and a quadratic (y = ax² + bx + c). Make sure both are in y = form.",
},
{
step: "2",
title: "Set the right sides equal",
body: "Since both equal y, set them equal to each other: mx + b = ax² + bx + c.",
},
{
step: "3",
title: "Rearrange to zero",
body: "Move everything to one side: 0 = ax² + (bm)x + (cb). You now have a new quadratic equation.",
},
{
step: "4",
title: "Use the Discriminant on the new quadratic",
body: "Δ > 0: line crosses parabola at 2 points. Δ = 0: line is tangent (1 point). Δ < 0: line misses parabola (0 points).",
},
].map((item) => (
<div
key={item.step}
className="flex gap-4 bg-white rounded-xl p-4 border border-violet-100"
>
<div className="w-8 h-8 bg-violet-600 text-white rounded-full flex items-center justify-center font-bold shrink-0">
{item.step}
</div>
<div>
<p className="font-bold text-slate-800 mb-1">
{item.title}
</p>
<p className="text-slate-600 text-sm">{item.body}</p>
</div>
</div>
))}
</div>
<div className="bg-violet-100 rounded-xl p-5">
<p className="font-bold text-violet-900 mb-2">Worked Example</p>
<p className="text-sm text-slate-700 mb-3">
Find all intersections of y = 2x + 1 and y = x² 2x + 3.
</p>
<div className="font-mono text-sm space-y-1 text-slate-600">
<p>Set equal: 2x + 1 = x² 2x + 3</p>
<p>Rearrange: 0 = x² 4x + 2</p>
<p>Discriminant: Δ = (4)² 4(1)(2) = 16 8 = 8</p>
<p className="text-violet-700 font-bold">
Δ &gt; 0 2 intersection points
</p>
<p>
x = <Frac n="4 ± √8" d="2" /> = 2 ± 2
</p>
</div>
</div>
<div className="bg-red-50 border border-red-200 rounded-xl p-4 text-sm">
<p className="font-bold text-red-800 mb-1">
Key Distinction: Tangent vs. Intersects
</p>
<p className="text-slate-700">
When the SAT says the line is <em>tangent</em> to the parabola,
that means exactly 1 intersection set Δ = 0 and solve for the
unknown constant.
</p>
</div>
</div>
<LinearQuadraticSystemWidget />
<button
onClick={() => scrollToSection(3)}
className="mt-12 group flex items-center text-violet-600 font-bold hover:text-violet-800 transition-colors"
>
Next: Practice Quiz{" "}
<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
</button>
</section>
{/* Section 4: Quiz */}
<section
ref={(el) => {
sectionsRef.current[3] = el;
}}
className="min-h-screen flex flex-col justify-center"
>
<h2 className="text-4xl font-extrabold text-slate-900 mb-8">
Practice Time
</h2>
{QUADRATIC_EQ_QUIZ_DATA.map((quiz, idx) => (
<div key={quiz.id} className="mb-12">
<Quiz data={quiz} />
</div>
))}
<div className="p-8 bg-violet-900 rounded-2xl text-white text-center mt-12">
<h3 className="text-2xl font-bold mb-4">Topic Mastered!</h3>
<button
onClick={onFinish}
className="px-6 py-3 bg-white text-violet-900 font-bold rounded-full hover:bg-violet-50 transition-colors"
>
Finish Lesson
</button>
</div>
</section>
</div>
</div>
);
};
export default QuadraticEquationsLesson;