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<!DOCTYPE html>
<html data-require="math">
<head>
<meta charset="UTF-8" />
<title>Factoring difference of squares 3</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars" data-ensure="getGCD( A, B ) === 1">
<var id="A">randRange( 2, 6 )</var>
<var id="B">randRange( 1, 6 )</var>
<var id="F">randFromArray([2, 3, 5])</var>
<var id="SQUARE">A * A</var>
<var id="CONSTANT">-B * B</var>
<var id="TERM1">F</var>
<var id="TERM1N">"[-\\u2212]" + F</var>
<var id="TERM2">"\\(\\s*" + A + "\\s*[xX]\\s*\\+\\s*" + B + "\\s*\\)"</var>
<var id="TERM2N">"\\(\\s*[-\\u2212]" + A + "\\s*[xX]\\s*[-\\u2212]\\s*" + B + "\\s*\\)"</var>
<var id="TERM3">"\\(\\s*" + A + "\\s*[xX]\\s*[-\\u2212]\\s*" + B + "\\s*\\)"</var>
<var id="TERM3N">"\\(\\s*[-\\u2212]" + A + "\\s*[xX]\\s*\\+\\s*" + B + "\\s*\\)"</var>
</div>
<div>
<p class="question">
Factor the following expression:
</p>
<p>
<code><var>F * SQUARE</var>x^2 + <var>F * CONSTANT</var></code>
</p>
</div>
<div class="solution" data-type="set">
<!-- the three terms can be in any order, and any 2 of them can be negated -->
<div class="set-sol" data-type="regex">^\s*<var>TERM1</var>\s*<var>TERM2</var>\s*<var>TERM3</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1</var>\s*<var>TERM3</var>\s*<var>TERM2</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2</var>\s*<var>TERM1</var>\s*<var>TERM3</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2</var>\s*<var>TERM3</var>\s*<var>TERM1</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3</var>\s*<var>TERM1</var>\s*<var>TERM2</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3</var>\s*<var>TERM2</var>\s*<var>TERM1</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1N</var>\s*<var>TERM2N</var>\s*<var>TERM3</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1N</var>\s*<var>TERM3N</var>\s*<var>TERM2</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2N</var>\s*<var>TERM1N</var>\s*<var>TERM3</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2N</var>\s*<var>TERM3N</var>\s*<var>TERM1</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3N</var>\s*<var>TERM1N</var>\s*<var>TERM2</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3N</var>\s*<var>TERM2N</var>\s*<var>TERM1</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1N</var>\s*<var>TERM2</var>\s*<var>TERM3N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1N</var>\s*<var>TERM3</var>\s*<var>TERM2N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2N</var>\s*<var>TERM1</var>\s*<var>TERM3N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2N</var>\s*<var>TERM3</var>\s*<var>TERM1N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3N</var>\s*<var>TERM1</var>\s*<var>TERM2N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3N</var>\s*<var>TERM2</var>\s*<var>TERM1N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1</var>\s*<var>TERM2N</var>\s*<var>TERM3N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM1</var>\s*<var>TERM3N</var>\s*<var>TERM2N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2</var>\s*<var>TERM1N</var>\s*<var>TERM3N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM2</var>\s*<var>TERM3N</var>\s*<var>TERM1N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3</var>\s*<var>TERM1N</var>\s*<var>TERM2N</var>\s*$</div>
<div class="set-sol" data-type="regex">^\s*<var>TERM3</var>\s*<var>TERM2N</var>\s*<var>TERM1N</var>\s*$</div>
<div class="input-format"><span class="entry"></span></div>
<div class="example">a factored expression, like <b>2(3x+1)(3x+2)</b></div>
</div>
<div class="hints">
<div>
<p>We can start by factoring a <code>\green{<var>F</var>}</code> out of each term:</p>
<code>\qquad \green{<var>F</var>}(\pink{<var>SQUARE</var>x^2} - \blue{<var>abs(CONSTANT)</var>})</code>
</div>
<p>
The second term is of the form <code>\color{<var>PINK</var>}{a^2} - \color{<var>BLUE</var>}{b^2}</code>,
which is a difference of two squares so we can factor it as
<code>\green{<var>F</var>}(\pink{a} + \blue{b})
(\color{<var>PINK</var>}{a} - \color{<var>BLUE</var>}{b})</code>.
</p>
<p>
What are the values of <code class="hint_pink">a</code> and <code class="hint_blue">b</code>?
</p>
<div>
<p><code class="hint_pink">\qquad a = \sqrt{<var>SQUARE</var>x^2} = <var>A</var>x</code></p>
<p><code class="hint_blue">\qquad b = \sqrt{<var>B * B</var>} = <var>B</var></code></p>
</div>
<p>
Use the values we found for <code class="hint_pink">a</code> and <code class="hint_blue">b</code>
to complete the factored expression,
<code>\green{<var>F</var>}(\color{<var>PINK</var>}{a} + \color{<var>BLUE</var>}{b})
(\color{<var>PINK</var>}{a} - \color{<var>BLUE</var>}{b})</code>.
</p>
<p class="final_answer">
So we can factor the expression as:
<code>\green{<var>F</var>}(\color{<var>PINK</var>}{<var>A</var>x} + \color{<var>BLUE</var>}{<var>B</var>})
(\color{<var>PINK</var>}{<var>A</var>x} - \color{<var>BLUE</var>}{<var>B</var>})</code>
</p>
</div>
</div>
</div>
</div>
</body>
</html>
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