<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>AlgMap-15.2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[Solve the following quadratic equation by completing the square:
\[ {@v^2+a*v+b@} = 0. \] 
<br>
Give the completed square form: [[input:cans]] [[validation:cans]]\(=0.\)
<br>
<br>
Hence solve the quadratic, giving your answer as a set \(\left(\mbox e.g.\ \left\{x_1 , x_2\right\} \right)\): <br>
<br>
\({@v@} \in \) [[input:ans]] [[validation:ans]][[feedback:prt1]][[feedback:comp]]]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text>When solving a more complicated quadratic a better option, than factorising, is completing the square (The method by which we remove the {@v@} term). The general rule (when the \(x^2\) coeffecient is one) is:

\[ x^2+ax+b = \left(x+\frac{a}{2} \right)^2-\frac{a^2}{4}+c. \]

Once you have removed the \(x\) term you can rearrange the formula to find \((x+\frac{a}{2})^2\). Then you simply need to square root and rearrange to find \(x\).

In this example:

\[ {@v^2+a*v+b@} = 0, \]

we complete the square:

\[ {@(v+a/2)^2-a^2/4+b@} = 0. \]

Rearranging gives:

\[ {@(v+a/2)^2@} = {@a^2/4-b@}. \]

Square root (remember to take plus and minus):

\[ {@(v+a/2)@} = \pm {@sqrt(a^2/4-b)@}. \]

Finally:

\[ {@v@} = {@-a/2@} \pm {@sqrt(a^2/4-b)@}. \]

Therefeore, our solution set is:

\[ {@v@} \in \left\{ {@-a/2+sqrt(a^2/4-b)@} , {@-a/2-sqrt(a^2/4-b)@}    \right\}. \]</text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>v:rand([x,y,z,t]);
a:rand(5)+5;
b:rand(5)+1;
ta1:-a/2+sqrt(a^2/4-b);
ta2:-a/2-sqrt(a^2/4-b);
cta:(v+a/2)^2-a^2/4+b;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[ {@v^2+a*v+b@} = {@cta@} = 0. \] 
\[ {@v@} \in \left\{ {@-a/2+sqrt(a^2/4-b)@} , {@-a/2-sqrt(a^2/4-b)@}    \right\}. \]</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>none</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans</name>
      <type>algebraic</type>
      <tans>{ta1,ta2}</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>1</showvalidation>
      <options/>
    </input>
    <input>
      <name>cans</name>
      <type>algebraic</type>
      <tans>cta</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>1</showvalidation>
      <options/>
    </input>
    <prt>
      <name>comp</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>CompSquare</answertest>
        <sans>cans</sans>
        <tans>cta</tans>
        <testoptions>v</testoptions>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>comp-0-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>comp-0-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt1</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans</sans>
        <tans>{ta1, ta2 }</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>1-0-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>1</falsenextnode>
        <falseanswernote>1-0-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans</sans>
        <tans>{?,?}</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>1-1-T </trueanswernote>
        <truefeedback format="html">
          <text>Replace the question marks with the solutions to the quadratic.</text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>1-1-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>741178775</deployedseed>
    <deployedseed>2138356284</deployedseed>
    <deployedseed>1495657118</deployedseed>
    <deployedseed>645275494</deployedseed>
    <deployedseed>89311231</deployedseed>
    <deployedseed>1974882855</deployedseed>
    <deployedseed>1322410172</deployedseed>
    <deployedseed>1619753723</deployedseed>
    <deployedseed>1728122350</deployedseed>
    <deployedseed>1297765813</deployedseed>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans</name>
        <value>{ta1, ta2}</value>
      </testinput>
      <testinput>
        <name>cans</name>
        <value>cta</value>
      </testinput>
      <expected>
        <name>comp</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>comp-0-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>1-0-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans</name>
        <value>{?,?}</value>
      </testinput>
      <testinput>
        <name>cans</name>
        <value>v^2+a*v+b</value>
      </testinput>
      <expected>
        <name>comp</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>comp-0-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt1</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
