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<title>Factoring difference of squares 3</title>
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<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>
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<p class="question">
Factor the expression below completely. All coefficients should be integers.
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<p class="problem"><code><var>F * SQUARE</var>x^2 + <var>F * CONSTANT</var></code></p>
<div class="solution" data-type="expression" data-same-form>
<var>F</var>(<var>A</var>x + <var>B</var>)(<var>A</var>x - <var>B</var>)
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<div class="hints">
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<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>
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<p>
The second term is of the form <code>\pink{a^2} - \blue{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})(\pink{a} - \blue{b})</code>.
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<p>What are the values of <code>\pink{a}</code> and <code>\blue{b}</code>?</p>
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<p><code>\qquad \pink{a = \sqrt{<var>SQUARE</var>x^2} = <var>A</var>x}</code></p>
<p><code>\qquad \blue{b = \sqrt{<var>B * B</var>} = <var>B</var>}</code></p>
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<p>
Use the values we found for <code>\pink{a}</code> and <code>\blue{b}</code>
to complete the factored expression,
<code>\green{<var>F</var>}(\pink{a} + \blue{b})(\pink{a} - \blue{b})</code>.
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<p>
So we can factor the expression as:
<code>\green{<var>F</var>}(\pink{<var>A</var>x} + \blue{<var>B</var>})
(\pink{<var>A</var>x} - \blue{<var>B</var>})</code>
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