// Numbas version: exam_results_page_options {"name": "Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Factorise a quadratic equation where the coefficient of the $x^2$ term is greater than 1 and then write down the roots of the equation

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$d$ in $(ax+b)(cx+d)$

"}, "a1": {"group": "last q", "name": "a1", "definition": "random(-3..-1)", "templateType": "anything", "description": ""}, "a": {"group": "last q", "name": "a", "definition": "random(2..3)", "templateType": "anything", "description": "

$a$ in $(ax+b)(cx+d)$

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$b$ in $(ax+b)(cx+d)$

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$c$ in $(ax+b)(cx+d)$

"}}, "name": "Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1", "ungrouped_variables": [], "variablesTest": {"maxRuns": "150", "condition": "coprime(a,b) and coprime(c,d) and (a*d <> -b*c)"}, "advice": "

a)

As this question involves a number greater than $1$ before the $x^2$ value it has a factorised form $(ax+b)(cx+d)$.

To find $a$ and $c$, we need to consider the factors of $\\var{a*c}$.

We are already given that one of them is $\\var{a}$, so we know that the other one must be $\\var{c}$.

This means our factorised equation must take the form

\\[(\\var{a}x+b)(\\var{c}x+d)=0\\text{.}\\]

This expands to

\\[ \\simplify{ {a*c}x^2 + ({a}*d+{c}*b)x + a*b} \\]

So we must find two numbers which add together to make $\\var{a*d+b*c}$, and multiply together to make $\\var{b*d}$.

Therefore $b$ and $d$ must satisfy

\\begin{align}
b \\times d &=\\var{b*d}\\\\
\\simplify{{a}d+{c}b} &= \\var{a*d+b*c}\\text{.}
\\end{align}

$b = \\var{b}$ and $d = \\var{d}$ satisfy these equations:

\\begin{align}
\\var{b} \\times \\var{d} &=\\var{b*d}\\\\
\\simplify[]{ {a}*{d} + {b}*{c} } &= \\var{a*d+b*c}
\\end{align}

So the factorised form of the equation is

\\[ \\simplify{({a}x+{b})({c}x+{d}) = 0} \\text{.}\\]

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Factorise the expression:

$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}}\\text{.}$

Factorise the expression:

$ \\simplify{ {d1*b1} + {a1*d1 + b1*c1}x + {a1*c1}x^2 } $

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