474 lines
20 KiB
TypeScript
474 lines
20 KiB
TypeScript
import React, { useRef, useState, useEffect } from "react";
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import { ArrowDown, Check, BookOpen, Target, Grid } from "lucide-react";
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import RationalExplorer from "../../../components/lessons/RationalExplorer";
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import RadicalSolutionWidget from "../../../components/lessons/RadicalSolutionWidget";
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import Quiz from "../../../components/lessons/Quiz";
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import { ADV_RATIONAL_QUIZ } from "../../../utils/constants";
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import { Frac } from "../../../components/Math";
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interface LessonProps {
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onFinish?: () => void;
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}
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const RationalRadicalLesson: React.FC<LessonProps> = ({ onFinish }) => {
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const [activeSection, setActiveSection] = useState(0);
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const sectionsRef = useRef<(HTMLElement | null)[]>([]);
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const scrollToSection = (index: number) => {
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setActiveSection(index);
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sectionsRef.current[index]?.scrollIntoView({
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behavior: "smooth",
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block: "start",
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});
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};
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useEffect(() => {
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const observer = new IntersectionObserver(
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(entries) => {
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entries.forEach((entry) => {
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if (entry.isIntersecting) {
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const index = sectionsRef.current.indexOf(
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entry.target as HTMLElement,
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);
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if (index !== -1) setActiveSection(index);
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}
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});
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},
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{ rootMargin: "-20% 0px -60% 0px" },
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);
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sectionsRef.current.forEach((section) => {
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if (section) observer.observe(section);
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});
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return () => observer.disconnect();
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}, []);
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const SectionMarker = ({
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index,
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title,
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icon: Icon,
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}: {
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index: number;
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title: string;
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icon: any;
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}) => {
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const isActive = activeSection === index;
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const isPast = activeSection > index;
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return (
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<button
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onClick={() => scrollToSection(index)}
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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"}`}
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>
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<div
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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"}`}
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>
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{isPast ? (
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<Check className="w-4 h-4" />
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) : (
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<Icon className="w-4 h-4" />
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)}
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</div>
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<div className="text-left">
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<p
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className={`text-sm font-bold ${isActive ? "text-violet-900" : "text-slate-600"}`}
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>
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{title}
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</p>
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</div>
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</button>
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);
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};
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return (
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<div className="flex flex-col lg:flex-row min-h-screen">
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<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 lg:bg-transparent z-0 hidden lg:block">
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<nav className="space-y-2">
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<SectionMarker index={0} title="Rational Functions" icon={Grid} />
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<SectionMarker index={1} title="Radical Equations" icon={Target} />
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<SectionMarker index={2} title="Practice" icon={BookOpen} />
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</nav>
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</aside>
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<div className="flex-1 lg:ml-64 p-6 md:p-12 max-w-4xl mx-auto">
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{/* Section 1: Rational Functions */}
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<section
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ref={(el) => {
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sectionsRef.current[0] = el;
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}}
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className="min-h-screen flex flex-col justify-center mb-24 pt-20 lg:pt-0"
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>
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<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
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Rational Functions & Discontinuities
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</h2>
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<div className="prose prose-slate text-lg text-slate-600 mb-8">
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<p>
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A <strong>rational function</strong> is any function of the form
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f(x) = P(x) ÷ Q(x), where both P and Q are polynomials. Wherever
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Q(x) = 0, the function is undefined — creating a{" "}
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<strong>discontinuity</strong>. There are two types of
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discontinuities: holes and vertical asymptotes.
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</p>
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</div>
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{/* Holes vs Vertical Asymptotes */}
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<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-8 space-y-5">
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<h3 className="text-lg font-bold text-violet-900">
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Holes vs. Vertical Asymptotes
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</h3>
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<div className="bg-violet-100 rounded-xl p-4 text-sm">
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<p className="font-bold text-violet-900 mb-1">
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Key Step: Always Factor Both Numerator and Denominator First
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</p>
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<p className="text-slate-700">
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Once factored, look at the denominator's zeros. If a zero also
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cancels with the numerator → <strong>hole</strong>. If it does
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not cancel → <strong>vertical asymptote</strong>.
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</p>
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</div>
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<div className="grid grid-cols-1 md:grid-cols-2 gap-4">
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<div className="text-center mb-3">
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<span className="inline-block bg-amber-100 text-amber-800 font-bold px-3 py-1 rounded-full text-sm">
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Hole (Removable Discontinuity)
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</span>
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</div>
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<p className="text-slate-600 text-sm mb-2">
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Occurs when a factor in the denominator{" "}
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<strong>also cancels</strong> with a factor in the numerator.
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The function is undefined at that x-value, but there's no
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asymptote.
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</p>
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<div className="bg-amber-50 rounded-lg p-3 font-mono text-sm">
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<p className="text-slate-600">
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f(x) = <Frac n="(x − 2)(x + 3)" d="(x − 2)(x − 1)" />
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</p>
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<p className="text-slate-600">
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After canceling: f(x) = <Frac n="x + 3" d="x − 1" />
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</p>
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<p className="text-amber-700 font-bold">
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Hole at x = 2 (canceled factor)
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</p>
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<p className="text-violet-700 font-bold">
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V. asymptote at x = 1 (remains)
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</p>
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</div>
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<p className="text-slate-500 text-xs mt-2">
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To find the y-coordinate of the hole: plug x = 2 into the
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simplified function: f(2) = (2 + 3) ÷ (2 − 1) = 5. Hole is at
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(2, 5).
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</p>
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</div>
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<div className="text-center mb-3">
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<span className="inline-block bg-red-100 text-red-800 font-bold px-3 py-1 rounded-full text-sm">
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Vertical Asymptote
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</span>
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</div>
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<p className="text-slate-600 text-sm mb-2">
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Occurs when a denominator factor does{" "}
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<strong>NOT cancel</strong>. The function approaches ±∞ near
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that x-value. The graph has a vertical line it never crosses.
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</p>
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<div className="bg-red-50 rounded-lg p-3 font-mono text-sm">
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<p className="text-slate-600">
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f(x) = <Frac n="x + 1" d="(x − 3)(x + 2)" />
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</p>
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<p className="text-slate-600">No factors cancel.</p>
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<p className="text-red-700 font-bold">
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V. asymptotes at x = 3 and x = −2
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</p>
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</div>
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<p className="text-slate-500 text-xs mt-2">
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The denominator has zeros at x = 3 and x = −2. Neither cancels
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with the numerator, so both are vertical asymptotes.
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</p>
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</div>
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</div>
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{/* Horizontal Asymptotes */}
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<p className="font-bold text-violet-800 mb-3">
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Horizontal Asymptotes — End Behavior of Rational Functions
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</p>
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<p className="text-slate-600 text-sm mb-3">
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Compare the degree of the numerator (n) to the degree of the
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denominator (d):
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</p>
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<div className="overflow-x-auto">
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<table className="w-full text-sm border-collapse">
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<thead>
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<tr className="bg-violet-100 text-violet-900">
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<th className="p-2 text-left font-bold">Condition</th>
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<th className="p-2 text-left font-bold">
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Horizontal Asymptote
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</th>
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</tr>
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</thead>
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<tbody className="text-slate-600">
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<tr className="border-b border-violet-50">
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<td className="p-2 font-semibold">
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n < d (numerator degree lower)
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</td>
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<td className="p-2">y = 0</td>
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</tr>
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<tr className="border-b border-violet-50 bg-violet-50">
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<td className="p-2 font-semibold">n = d (same degree)</td>
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<td className="p-2">
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y ={" "}
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<Frac
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n="leading coefficient of top"
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d="leading coefficient of bottom"
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/>
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</td>
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</tr>
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<tr>
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<td className="p-2 font-semibold">
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n > d (numerator degree higher)
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</td>
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<td className="p-2">
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No horizontal asymptote (oblique asymptote instead)
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</td>
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</tr>
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</tbody>
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</table>
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</div>
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<div className="mt-3 bg-violet-50 rounded-lg p-3 text-sm font-mono">
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<p className="text-slate-600">
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f(x) = <Frac n="3x²" d="x² − 4" />: n = d = 2 → H.A. at y ={" "}
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<Frac n="3" d="1" /> ={" "}
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<strong className="text-violet-700">3</strong>
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</p>
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</div>
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</div>
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</div>
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<RationalExplorer />
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<button
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onClick={() => scrollToSection(1)}
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className="mt-12 group flex items-center text-violet-600 font-bold hover:text-violet-800 transition-colors"
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>
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Next: Radical Equations{" "}
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<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
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</button>
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</section>
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{/* Section 2: Radical Equations */}
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<section
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ref={(el) => {
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sectionsRef.current[1] = el;
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}}
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className="min-h-screen flex flex-col justify-center mb-24"
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>
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<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
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Radical Equations & Extraneous Solutions
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</h2>
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<div className="prose prose-slate text-lg text-slate-600 mb-8">
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<p>
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A <strong>radical equation</strong> has the variable under a
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square root (or other radical). The key strategy is to isolate the
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radical and then raise both sides to the appropriate power to
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eliminate it. This process can introduce{" "}
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<strong>extraneous solutions</strong> — values that satisfy the
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transformed equation but not the original. You{" "}
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<strong>must always check</strong> your answers in the original
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equation.
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</p>
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</div>
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<div className="bg-violet-50 border border-violet-200 rounded-2xl p-6 mb-8 space-y-5">
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<h3 className="text-lg font-bold text-violet-900">
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Solving Radical Equations: Step-by-Step
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</h3>
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<p className="font-bold text-violet-800 mb-3">General Process</p>
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<div className="space-y-2">
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{[
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{
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step: "1",
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text: "Isolate the radical on one side of the equation.",
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},
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{
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step: "2",
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text: "Square both sides (for square roots) or cube both sides (for cube roots).",
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},
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{
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step: "3",
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text: "Solve the resulting equation (may be linear or quadratic).",
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},
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{
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step: "4",
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text: "CHECK all solutions in the ORIGINAL equation. Reject any that make the original false.",
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},
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].map((item) => (
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<div key={item.step} className="flex gap-3 items-start">
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<div className="w-6 h-6 bg-violet-600 text-white rounded-full flex items-center justify-center font-bold text-xs shrink-0">
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{item.step}
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</div>
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<p className="text-slate-600 text-sm">{item.text}</p>
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</div>
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))}
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</div>
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</div>
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<div className="grid grid-cols-1 md:grid-cols-2 gap-4">
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<p className="font-bold text-green-700 mb-3">
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Example 1 — No Extraneous Solution
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</p>
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<div className="font-mono text-sm space-y-1 text-slate-600">
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<p>√(x + 3) = 5</p>
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<p>Square both sides:</p>
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<p>x + 3 = 25</p>
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<p>x = 22</p>
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<p className="text-slate-500 mt-1">
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Check: √(22 + 3) = √25 = 5 ✓
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</p>
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<p className="text-green-700 font-bold">x = 22 ✓ (valid)</p>
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</div>
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</div>
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<p className="font-bold text-red-700 mb-3">
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Example 2 — Extraneous Solution Found!
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</p>
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<div className="font-mono text-sm space-y-1 text-slate-600">
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<p>√(2x + 1) = x − 1</p>
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<p>Square both sides:</p>
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<p>2x + 1 = x² − 2x + 1</p>
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<p>0 = x² − 4x → x(x − 4) = 0</p>
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<p>x = 0 or x = 4</p>
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<p className="text-slate-500 mt-1">
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Check x = 0: √1 = 0 − 1 → 1 ≠ −1 ✗
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</p>
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<p className="text-slate-500">
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Check x = 4: √9 = 4 − 1 → 3 = 3 ✓
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</p>
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<p className="text-red-700 font-bold">
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x = 0 is extraneous! Only x = 4.
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</p>
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</div>
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</div>
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</div>
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{/* Why Extraneous Solutions Occur */}
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<div className="bg-violet-100 rounded-xl p-4 text-sm">
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<p className="font-bold text-violet-900 mb-1">
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Why Do Extraneous Solutions Appear?
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</p>
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<p className="text-slate-700">
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Squaring both sides is not a reversible step — if you square a
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negative number, it becomes positive. For example, if the
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original equation has √x = A, then A must be non-negative (a
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square root can never give a negative output). But squaring
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produces solutions for both A and −A. Any solution that would
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require the radical to equal a negative number is extraneous.
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</p>
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</div>
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{/* Rational Exponents */}
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<div className="bg-white rounded-xl p-5 border border-violet-100">
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<p className="font-bold text-violet-800 mb-3">
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Rational Exponents: Connecting Radicals and Powers
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</p>
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<div className="bg-violet-50 rounded-lg p-3 text-center mb-3">
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<p className="font-mono text-violet-800 font-bold text-lg">
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x<sup>m/n</sup> = (<sup>n</sup>√x)<sup>m</sup> = <sup>n</sup>
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√(x<sup>m</sup>)
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</p>
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</div>
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<div className="grid grid-cols-1 sm:grid-cols-3 gap-3 text-sm">
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<div className="bg-violet-50 rounded-lg p-3 text-center">
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<p className="font-mono text-violet-700 font-bold">
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x<sup>1/2</sup> = √x
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</p>
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<p className="text-slate-500 text-xs mt-1">
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Exponent ½ = square root
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</p>
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</div>
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<div className="bg-violet-50 rounded-lg p-3 text-center">
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<p className="font-mono text-violet-700 font-bold">
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x<sup>1/3</sup> = <sup>3</sup>√x
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</p>
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<p className="text-slate-500 text-xs mt-1">
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Exponent ⅓ = cube root
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</p>
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</div>
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<div className="bg-violet-50 rounded-lg p-3 text-center">
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<p className="font-mono text-violet-700 font-bold">
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x<sup>2/3</sup> = (<sup>3</sup>√x)²
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</p>
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<p className="text-slate-500 text-xs mt-1">
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Numerator = power, denominator = root
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</p>
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</div>
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</div>
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<div className="mt-3 bg-violet-50 rounded-lg p-3 font-mono text-sm">
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<p className="text-slate-600">
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Solve: x<sup>2/3</sup> = 4
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</p>
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<p className="text-slate-600">
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Raise both sides to power 3/2: x = 4<sup>3/2</sup> = (√4)³ =
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2³ = <strong className="text-violet-700">8</strong>
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</p>
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</div>
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</div>
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<div className="bg-red-50 border border-red-200 rounded-xl p-4 text-sm">
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<p className="font-bold text-red-800 mb-1">
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SAT Trap: The Domain of Radical Functions
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</p>
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<p className="text-slate-700">
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For even roots (square root, fourth root, etc.), the expression
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under the radical must be <strong>≥ 0</strong>. The SAT may test
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this by asking for the domain of f(x) = √(2x − 6). Set 2x − 6 ≥
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0 → x ≥ 3. The domain is x ≥ 3.
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</p>
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</div>
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</div>
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<RadicalSolutionWidget />
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<button
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onClick={() => scrollToSection(2)}
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className="mt-12 group flex items-center text-violet-600 font-bold hover:text-violet-800 transition-colors"
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>
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Next: Practice Quiz{" "}
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<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
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</button>
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</section>
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{/* Section 3: Quiz */}
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<section
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ref={(el) => {
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sectionsRef.current[2] = el;
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}}
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className="min-h-screen flex flex-col justify-center"
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>
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<h2 className="text-4xl font-extrabold text-slate-900 mb-8">
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Practice Time
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</h2>
|
||
{ADV_RATIONAL_QUIZ.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 RationalRadicalLesson;
|