// Numbas version: exam_results_page_options {"name": "Simon's copy of Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "name": "Simon's copy of Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1", "advice": "

#### (a)

\n

This question is more difficult because we are factorising a quadratic of the form $ax^2+bx+c$ where $a \\neq 1$.

\n

To factorise an expression like $ax^2+bx+c$ we can start by finding two numbers which multiply to give $a\\times c$ and add to give $b$.

\n

\n

\n

In our example, to factorise $\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}$ we start by finding two numbers which:

\n

Multiply to give $\\var{a*b*c*d}$   $(a \\times c)$ and add to give $\\simplify{{a}*{d}+{b}*{c}}$    $(b)$

\n

\n

\n

Note that $(\\var{a*d}) \\times (\\var{b*c}) = \\var{a*b*c*d}$

\n

and

\n

($\\var{a*d})+(\\var{b*c}) = \\var{a*d+b*c}$

\n

So the two numbers required are $\\var{a*d}$ and $\\var{b*c}$.

\n

\n

We can then return to our original quadratic and split the $x$ coefficient into two pieces, $\\var{a*d}x$ and $\\var{b*c}x$, as follows:

\n

$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}}= \\simplify[!cancelterms]{{a*c}x^2+{a*d}x+{b*c}x+{b*d}}$

\n

\n

Now we can factorise the first two terms and the last two terms separately as:

\n

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

\n

which is the same as

\n

$\\simplify[!cancelterms]{({a}x+{b})({c}x+{d})}$ which is thus our final factorisation.

\n

\n

\n

\n

\n

#### (b)

\n

$\\simplify{({a}x+{b})({c}x+{d}) = 0}$ when either $\\simplify{{a}x+{b}} = 0$  or  $\\simplify{{c}x+ {d}} = 0$.

\n

So the roots of the equation are $\\var[fractionnumbers]{-b/a}$ and $\\var[fractionnumbers]{-d/c}$.

\n

", "variables": {"d": {"templateType": "anything", "name": "d", "description": "

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

", "group": "last q", "definition": "random(-6..6 except 0)"}, "c": {"templateType": "anything", "name": "c", "description": "

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

", "group": "last q", "definition": "random(2..6 except a)"}, "b": {"templateType": "anything", "name": "b", "description": "

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

", "group": "last q", "definition": "random(-4..4 except 0)"}, "a": {"templateType": "anything", "name": "a", "description": "

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

", "group": "last q", "definition": "random(2..3)"}, "roots": {"templateType": "anything", "name": "roots", "description": "

The roots of the equation

", "group": "last q", "definition": "sort([-b/a,-d/c])"}}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": [], "parts": [{"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "gaps": [{"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "allowFractions": false, "correctAnswerStyle": "plain", "minValue": "b", "type": "numberentry", "scripts": {}, "maxValue": "b", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerFraction": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"]}, {"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "allowFractions": false, "correctAnswerStyle": "plain", "minValue": "c", "type": "numberentry", "scripts": {}, "maxValue": "c", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerFraction": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"]}, {"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "allowFractions": false, "correctAnswerStyle": "plain", "minValue": "d", "type": "numberentry", "scripts": {}, "maxValue": "d", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerFraction": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"]}], "type": "gapfill", "prompt": "

Factorise the equation

\n

$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}\\text{.}$

\n

$(\\var{a}x+\\phantom{.}$[[0]]$) ($[[1]]$x+\\phantom{.}$[[2]]$)\\; = 0$

", "marks": 0, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}, {"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "gaps": [{"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "allowFractions": true, "correctAnswerStyle": "plain", "minValue": "roots[0]", "type": "numberentry", "scripts": {}, "maxValue": "roots[0]", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerFraction": true, "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"]}, {"variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "allowFractions": true, "correctAnswerStyle": "plain", "minValue": "roots[1]", "type": "numberentry", "scripts": {}, "maxValue": "roots[1]", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "correctAnswerFraction": true, "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"]}], "type": "gapfill", "prompt": "

\n

Write down the roots of the equation above.

\n

Input your answer as $x_1$ and $x_2$, where $x_1<x_2$.

\n

$x_1=$ [[0]]

\n

$x_2=$ [[1]]

", "marks": 0, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "rulesets": {}, "statement": "", "extensions": [], "variable_groups": [{"name": "last q", "variables": ["a", "b", "c", "d", "roots"]}], "functions": {}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}