// 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": [{"advice": "<h4>a)</h4>\n<p>As this question involves a number&nbsp;greater than $1$ before the $x^2$ value&nbsp;it has a factorised form $(ax+b)(cx+d)$.</p>\n<p>To find $a$ and $c$, we need to consider the factors of $\\var{a*c}$.</p>\n<p>We are already given that one of them is $\\var{a}$, so we know that the other one must be $\\var{c}$.</p>\n<p>This means our factorised equation must take the form</p>\n<p>\\[(\\var{a}x+b)(\\var{c}x+d)=0\\text{.}\\]</p>\n<p>This expands to</p>\n<p>\\[ \\simplify{ {a*c}x^2 + ({a}*d+{c}*b)x + a*b} \\]</p>\n<p>So we must find two numbers which add together to make $\\var{a*d+b*c}$, and multiply together to make $\\var{b*d}$.</p>\n<p>Therefore $b$ and $d$ must satisfy</p>\n<p>\\begin{align}<br/>b \\times d &amp;=\\var{b*d}\\\\<br/>\\simplify{{a}d+{c}b} &amp;= \\var{a*d+b*c}\\text{.}<br/>\\end{align}</p>\n<p>$b = \\var{b}$ and $d = \\var{d}$ satisfy these equations:</p>\n<p>\\begin{align}<br/>\\var{b} \\times \\var{d} &amp;=\\var{b*d}\\\\<br/>\\simplify[]{ {a}*{d} + {b}*{c} } &amp;= \\var{a*d+b*c}<br/>\\end{align}</p>\n<p>So the factorised form of the equation is&nbsp;</p>\n<p>\\[ \\simplify{({a}x+{b})({c}x+{d}) = 0} \\text{.}\\]</p>\n<h4>b)</h4>\n<p>$\\simplify{({a}x+{b})({c}x+{d}) = 0}$ when either $\\var{a}x+\\var{b} = 0$ or $\\var{c}x+ \\var{d} = 0$.</p>\n<p>So the roots of the equation are $\\var[fractionnumbers]{-b/a}$ and $\\var[fractionnumbers]{-d/c}$.</p>\n<p></p>", "statement": "", "variables": {"b": {"templateType": "anything", "name": "b", "description": "<p>$b$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(-5..5 except 0)"}, "c": {"templateType": "anything", "name": "c", "description": "<p><strong></strong>$c$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(2..8 except a)"}, "a": {"templateType": "anything", "name": "a", "description": "<p>$a$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(2..3)"}, "roots": {"templateType": "anything", "name": "roots", "description": "<p>The roots of the equation</p>", "group": "last q", "definition": "sort([-b/a,-d/c])"}, "d": {"templateType": "anything", "name": "d", "description": "<p>$d$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(-8..8 except 0)"}}, "tags": ["coefficient of x^2 greater than 1", "Factorisation", "factorisation", "factorising", "factorising quadratic equations", "Factorising quadratic equations", "factorising quadratic equations with x^2 coefficients greater than 1", "taxonomy"], "ungrouped_variables": [], "functions": {}, "rulesets": {}, "name": "Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1", "metadata": {"description": "<p>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</p>", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": [], "type": "question", "variable_groups": [{"variables": ["a", "b", "c", "d", "roots"], "name": "last q"}], "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "b", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "b", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "c", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "c", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "d", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "d", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Factorise the equation</p>\n<p>$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}\\text{.}$</p>\n<p>$(\\var{a}x+\\phantom{.}$[[0]]$) ($[[1]]$x+\\phantom{.}$[[2]]$)\\; = 0$</p>", "variableReplacements": [], "showFeedbackIcon": true}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "roots[0]", "correctAnswerFraction": true, "allowFractions": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "roots[0]", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "roots[1]", "correctAnswerFraction": true, "allowFractions": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "roots[1]", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p></p>\n<p>Write down the roots of the equation above.</p>\n<p>Input your answer as $x_1$ and $x_2$, where $x_1&lt;x_2$.</p>\n<p>$x_1=$ [[0]]</p>\n<p>$x_2=$&nbsp;[[1]]</p>", "variableReplacements": [], "showFeedbackIcon": true}], "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/"}]}]}], "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/"}]}