// Numbas version: finer_feedback_settings
{"name": "AD4 Quadratics - factorise (a not 1)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "AD4 Quadratics - factorise (a not 1)", "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"], "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"}, "statement": "", "advice": "<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>You may have to test a a few different options before you find one that works. In this case $a$ and $c$ are $\\var{a}$ and $\\var{c}$.</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<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>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "last q", "definition": "random(-5..5 except 0)", "description": "<p>$b$ in $(ax+b)(cx+d)$</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "last q", "definition": "random(2..8 except a)", "description": "<p><strong></strong>$c$ in $(ax+b)(cx+d)$</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "last q", "definition": "random(2..3)", "description": "<p>$a$ in $(ax+b)(cx+d)$</p>", "templateType": "anything", "can_override": false}, "roots": {"name": "roots", "group": "last q", "definition": "sort([-b/a,-d/c])", "description": "<p>The roots of the equation</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "last q", "definition": "random(-8..8 except 0)", "description": "<p>$d$ in $(ax+b)(cx+d)$</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "last q", "variables": ["a", "b", "c", "d", "roots"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, 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