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<h3 id="title" class="align-center">Al Azhar 9th Grade Lesson Notes</h3>
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<a href="/about/author.html" class="sidebar-link link">About the Author</a>
<a href="/about/school.html" class="sidebar-link link">About School</a>
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<h4>Table of Contents</h4>
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<h1 class="banner-text" id="banner-txt">Quadratic Equation: The Introduction</h1>
<h3 class="banner-text" id="banner-txt">Chapter 2</h3>
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<div class="chip" title="Beware of mistakes. Please report to me (Altaf) if you caught a mistake here."><span>❓ Unchecked by Professionals</span></div>
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<section>
<h1 id="introduction">Introduction</h1>
In algebra, a quadratic equation is any equation that can be rearranged in standard form as where x
represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0.<sup>[<a
href="#reference-1" class="link-compact">1</a>]</sup>
<h2 id="introduction--formula">Formula / Format</h2>
<span class="math-font">ax<sup>2</sup> + bx + c = 0</span>
<p>
<span class="math-font">a, b, c</span> = known numbers, where a ≠ 0 <br>
<span class="math-font">x</span> = the unknown, or the root
</p>
<h2 id="introduction--exercise">Exercises</h2>
<h3>Finding the roots with factorisation.</h3>
<ol>
<li>
<span id="q1.1.1"></span>
<p>
<hr>
<span>To find the answer, we need to find the factors of c that is suitable for b and c. <a href="finding-suitable.html" class="link">What and how do I find the suitable factors?</a></span><br>
<span>The suitable factor for the equation are <b>2 and -3</b>. </span><br>
<span>Then, insert each numbers to this formula: <span id="q1.1.2"></span>, where <span
class="math-font">y</span> is the factor.</span><br><br>
<span id="q1.1.3"></span><br><span id="q1.1.31"></span><br>
<hr>
<strong>So, the answer is <span id="q1.1.4"></span></strong>
</p>
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var element2 = document.getElementById("q1.1.2");
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katex.render(String.raw`x^2 - 1x - 6 = 0`, element1, {
throwOnError: false
});
katex.render(String.raw`x + y`, element2, {
throwOnError: false
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katex.render(String.raw`
\begin{aligned}
x + 2 &= 0\\
x &= -2\\
\end{aligned}
`, element3, {
throwOnError: true
}); katex.render(String.raw`
\begin{aligned}
x - 3 &= 0\\
x &= 3
\end{aligned}
`, element31, {
throwOnError: true
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katex.render(String.raw`x = \{-2, 3\}`, element4, {
throwOnError: false
});
</script>
</li>
<li>
<p>
<span id="q1.2.1"></span><br>
<span><strong>Factors:</strong> <span class="math-font">2, 4</span></span><br>
<hr>
<span id="q1.2.2"></span><br>
<span id="q1.2.3"></span><br>
<hr>
<span id="q1.2.4"></span><br>
</p>
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katex.render(String.raw`x^2 + 6x + 8 = 0`, element1, {
throwOnError: false
});
katex.render(String.raw`
\begin{aligned}
x + 2 &= 0\\
x &= -2\\
\end{aligned}
`, element2, {
throwOnError: true
}); katex.render(String.raw`
\begin{aligned}
x + 4 &= 0\\
x &= 4
\end{aligned}
`, element3, {
throwOnError: true
});
katex.render(String.raw`x = \{-2, -4\}`, element4, {
throwOnError: false
});
</script>
</li>
</ol>
<h1 id="relation-algebra">Relation between Quadratic Equation and Algebraic Expressions</h1>
<p>This is an algebraic expression → <span id="m0"></span>. You can turn that into a quadratic equation.</p>
<span id="m1"></span>
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const m0 = document.getElementById("m0");
const m1 = document.getElementById("m1");
katex.render(String.raw`(x + 5)(x + 7)`, m0, {
throwOnError: false
});
katex.render(String.raw`
\begin{aligned}
(x + 5)(x + 7) &= x^2 + 5x + 7x + 35&\\
&= x^2 + 12x + 35&\larr\textnormal{quadratic equation 😱😱}
\end{aligned}
`, m1, {
throwOnError: true
});
</script>
<h1 id="all-about-roots">Quadratic Equation's Root Operations</h1>
<h2 id="finding-qe">Finding Quadratic Equation with Roots</h2>
<p>You can use the formula below to find a Quadratic Equation with 2 roots.</p>
<span id="m2"></span><span>, where <span class=math-font>x<sub>1, 2</sub></span> are the roots.</span><br>
<p>
<strong>Example:</strong><br><br>
<span id="m3"></span>
</p>
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const m2 = document.getElementById("m2");
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katex.render(String.raw`(x - x_1)(x - x_2) = 0`, m2, {
throwOnError: false
});
katex.render(String.raw`
\begin{aligned}
(x - x_1)(x - x_2) &= 0\\
(x - (-3))(x - 4) &= 0\\
(x + 3))(x - 4) &= 0\\
x^2 - 4x + 3x - 12 &= 0\\
x^2 - x - 12 &= 0
\end{aligned}
`, m3, {
throwOnError: true
});
</script>
<h2 id="operations">Quadratic Root's Mathematical Operations</h2>
<h3 id="operations--formulas">Formulas</h3>
<p id="m4"></p>
<p id="m5"></p>
<h3 id="operations--examples">Examples</h3>
<ol>
<li>
<p>Diketahui persamaan kuadrat <span id="q1"></span>, memiliki akar-akar persamaan kuadrat <span class="math-font">x<sub>1</sub></span> dan <span class="math-font">x<sub>2</sub></span>. Jika <span class="math-font">x<sub>1</sub> < x<sub>2</sub></span>, maka tentukan:
<ol type="a">
<li><span class="math-font">x<sub>1</sub> + x<sub>2</sub></span></li>
<li><span class="math-font">x<sub>1</sub> × x<sub>2</sub></span></li>
</ol>
</p>
<hr>
<ol type="a">
<li><span id="a1"></span></li>
<li><span id="a2"></span></li>
</ol>
</li>
</ol>
<script>
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katex.render(String.raw`x_1 + x_2 = \dfrac{-b}{a}`, m4, {
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katex.render(String.raw`x_1 \times x_2 = \dfrac{c}{a}`, m5, {
throwOnError: false
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katex.render(String.raw`2x^2 + 7x + 3 = 0`, q1, {
throwOnError: false
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katex.render(String.raw`
\begin{aligned}
x_1 + x_2 &= \dfrac{-b}{a}\\
&= \dfrac{-7}{2}\\
\end{aligned}
`, a1, {
throwOnError: true
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katex.render(String.raw`
\begin{aligned}
x_1 \times x_2 &= \dfrac{c}{a}\\
&= \dfrac{3}{2}\\
\end{aligned}
`, a2, {
throwOnError: true
});
</script>
<h1>References</h1>
<ol>
<li id="reference-1" tabindex="-1">Quadratic Equation - The English Wikipedia, <a
href="https://en.wikipedia.org/wiki/Quadratic_equation"
class="link">https://en.wikipedia.org/wiki/Quadratic_equation</a></li>
</li>
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