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solving_quadratics_by_factoring.html
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solving_quadratics_by_factoring.html
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<!DOCTYPE html>
<html data-require="math">
<head>
<title>Solving Quadratics by Factoring</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<div data-ensure="A !== B">
<var id="A">randRangeNonZero( -10, 10 )</var>
<var id="B">randRangeNonZero( -10, 10 )</var>
</div>
<var id="SQUARE">randRange(1,6)</var>
</div>
<div class="problems">
<div id="original">
<div class="vars">
<var id="CONSTANT">SQUARE*A*B</var>
<var id="SIMPLECONSTANT">A*B</var>
<var id="LINEAR">SQUARE*(-A-B)</var>
<var id="SIMPLELINEAR">-A-B</var>
</div>
<p class="problem">Determine where <code>f(x)</code> intersects the <code>x</code>-axis.</p>
<p class="question"><code><var>SQUARE</var>x^2 + <var>LINEAR</var>x + <var>CONSTANT</var></code></p>
<p class="solution"><code>x=<var>A</var></code> and <code>x=<var>B</var></code></p>
<ul class="choices" data-none="true">
<li><code>x=<var>-A</var></code> and <code>x=<var>-B</var></code></li>
<li><code>x=<var>B</var></code> and <code>x=<var>A+B</var></code></li>
<li><code>x=<var>A</var></code> only</li>
<li><code>x=<var>B</var></code> only</li>
<li><code>x=<var>SQUARE</var></code> and <code>x=<var>LINEAR</var></code></li>
</ul>
<div class="hints" data-apply="appendContents">
<div id="hint1">
<p>The two numbers <code><var>-A</var></code> and <code><var>-B</var></code> satisfy both conditions:</p>
<p><code><var>-A</var> + <var>-B</var> = <var>SIMPLELINEAR</var></code></p>
<p><code><var>-A</var> \cdot <var>-B</var> = <var>SIMPLECONSTANT</var></code></p>
</div>
<p id="hint2">So <code>(x + <var>-A</var>)(x + <var>-B</var>) = 0</code>.</p>
<p id="hint3"><code>x + <var>-A</var> = 0</code> or <code>x + <var>-B</var> = 0</code></p>
<p id="hint4">Thus, <code>x = <var>A</var></code> and <code>x = <var>B</var></code> are the solutions.</p>
</div>
</div>
<div id="one-root" data-type="original">
<div class="vars">
<var id="CONSTANT">SQUARE * A * A</var>
<var id="SIMPLECONSTANT">A * A</var>
<var id="LINEAR">SQUARE * ( -2 * A )</var>
<var id="SIMPLELINEAR">-2 * A</var>
</div>
<p class="problem">Determine where <code>f(x)</code> intersects the x-axis.</p>
<p class="question"><code><var>SQUARE</var>x^2 + <var>LINEAR</var>x + <var>CONSTANT</var></code></p>
<p class="solution"><code>x=<var>A</var></code> only</p>
<ul class="choices" data-none="true">
<li><code>x=<var>-A</var></code> and <code>x=<var>A</var></code></li>
<li><code>x=<var>2 * A</var></code> and <code>x=<var>-A</var></code></li>
<li><code>x=<var>-2 * A</var></code> and <code>x=<var>A</var></code></li>
<li><code>x=<var>-A</var></code> only</li>
</ul>
<div class="hints" data-apply="appendContents">
<div id="hint1">
<p>The number <code><var>-A</var></code> used twice satisfies both conditions:</p>
<p><code><var>-A</var> + <var>-A</var> = <var>SIMPLELINEAR</var></code></p>
<p><code><var>-A</var> \cdot <var>-A</var> = <var>SIMPLECONSTANT</var></code></p>
</div>
<p id="hint2">So <code>(x + <var>-A</var>)^2 = 0</code>.</p>
<p id="hint3"><code>x + <var>-A</var> = 0</code></p>
<p id="hint4">Thus, <code>x = <var>A</var></code> is the solution.</p>
</div>
</div>
</div>
<div class="hints">
<div>
<p>The function intersects the <code>x</code>-axis where <code>f(x) = 0</code>, so solve the equation:</p>
<p><code><var>SQUARE</var>x^2 + <var>LINEAR</var>x + <var>CONSTANT</var>=0</code></p>
</div>
<div data-if="SQUARE > 1">
<p>Dividing both sides by <var>SQUARE</var> gives:</p>
<p><code>x^2 + <var>SIMPLELINEAR</var>x + <var>SIMPLECONSTANT</var>=0</code></p>
</div>
<p>The coefficient on the <code>x</code> term is <code><var>SIMPLELINEAR</var></code> and the constant term is <code><var>SIMPLECONSTANT</var></code>, so we need to find two numbers that add up to <code><var>SIMPLELINEAR</var></code> and multiply to <code><var>SIMPLECONSTANT</var></code>.</p>
</div>
</div>
</body>
</html>