// Numbas version: finer_feedback_settings {"name": "1.3.5.1 Factorise a quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.3.5.1 Factorise a quadratic", "tags": [], "metadata": {"description": "

Factorise a quadratic, ax^2+bx+c. An exam can choose whether or not a=1. Part of HELM Book 1.3

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Practise these questions repeatedly until you can quickly, confidently and correctly factorise the expressions.

", "advice": "

$\\var{expr}=\\var{ans}$

You can solve this by inspection or:

Given a quadratic of the form $x^2+Bx+C$, find two numbers that multiply to give $C$ and add to give $B$

$C=\\var{c}=(\\var{c1})\\times(\\var{c2})$ and $B=\\var{b}=(\\var{c1})+(\\var{c2})$

so $\\var{expr}=\\var{ans}$

Method 1 (the HELM book method)

\\[\\begin{align*}(\\var{a})(\\var{expr})&=(\\var{a})(\\var{a}\\var{ve}^2)+(\\var{a})(\\var{b}\\var{ve})+(\\var{a})(\\var{c})\\\\ &=(\\var{a}\\var{ve})^2+(\\var{b})(\\var{a}\\var{ve})+ \\var{a*c}\\end{align*}\\]

Let $\\var{a}\\var{ve}=u$, so

\\[\\begin{align*}(\\var{a})(\\var{expr}) &= u^2+\\var{b}u+\\var{a*c}\\\\ &= \\left(u+(\\var{c1*p2})\\right)\\left(u+(\\var{c2*p1})\\right)\\\\&=(\\var{a}\\var{ve}+(\\var{c1*p2})(\\var{a}\\var{ve}+(\\var{c2*p1}))\\\\&=(\\var{a})(\\var{ans})\\end{align*}\\]

so \\[\\var{expr}=\\var{ans}\\]

Method 2 (the A-C method)

Given a quadratic of the form $Ax^2+Bx+C$, find two numbers that multiply to give $A\\times C$ and add to give $B$

$AC = (\\var{a})\\times(\\var{c}) = \\var{a*c} = (\\var{p1*c2})\\times(\\var{p2*c1})$ and $B = \\var{b}=(\\var{p1*c2})+(\\var{p2*c1})$

\\[\\begin{align*}\\var{expr}&=\\var{a}\\var{ve}^2 + (\\var{p1*c2})\\var{ve} + (\\var{p2*c1})\\var{ve} + (\\var{c})\\\\&=\\left[\\var{a}\\var{ve}^2 + (\\var{p1*c2})\\var{ve}\\right] + \\left[(\\var{p2*c1})\\var{ve} + (\\var{c})\\right]\\\\&=(\\var{p1}\\var{ve})(\\var{term2})+(\\var{c1})(\\var{term2})\\\\&=\\var{ans} \\end{align*}\\]

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Factorised expression ax^2+bx+c is (p1.x-c1)(p1.x-c2). Default is for a=1 half the time.

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(p1x+c1)(p2x+c2) = p1.p2x^2 + (c1.p2+c2.p1)x+c1.c2

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(p1x+c1)(p2x+c2) = p1.p2x^2 + (c1.p2+c2.p1)x+c1.c2 = ax^2+bx+c

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for display in the advice

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for display in the advice

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Factorise $\\var{expr}$.

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