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    <name>
      <text>PT.2.9 (Finding independent variable)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p>Given \(f(x) = {@func2@}\) and that \(f(c) = -{@aa*bb@}\) then give one possible value of \(c\).</p>
<p>\(c=\)[[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text><![CDATA[<p>To begin, we substitute \(c\) into the function. We know that \(f(c)\) evaluates to \(-{@aa*bb@}\) therefore we have
\[
c^2 -{@aa+bb@}c = -{@aa*bb@} \implies c^2 -{@aa+bb@}c + {@aa*bb@}  = 0.
\]
Notice that we now have a quadratic in standard form. Therefore we can solve the quadratic using whichever tecnique we would like from week 1. In this case the quadratic factorises as 
\[
c^2 -{@aa+bb@}c + {@aa*bb@} = (c-{@aa@})(c-{@bb@}).
\]
Hence either \(c = {@aa@}\) or \(c = {@bb@}\).
</p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2020081700</text>
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    <questionvariables>
      <text>aa: 1+rand(10);
bb: 1+ rand(10);

func1: (x-aa)*(x-bb);
func2: expand((x-aa)*(x-bb)) - aa*bb;
ta1: aa;
ta2: bb;</text>
    </questionvariables>
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      <text>[[feedback:prt1]]</text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>[{@ta1@},{@ta2@}]</text>
    </questionnote>
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      <text/>
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      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
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