// Numbas version: exam_results_page_options
{"name": "Quadratics: Factorisation 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratics: Factorisation 2", "tags": ["Category: Quadratics"], "metadata": {"description": "<p>Factorising a quadratic expression of the form $ax^2+bx+c$, where $a&gt;1$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Factorise the following quadratic expression:</p>\n<p>\\[ \\simplify[unitFactor]{{a}x^2+{b}x+{c}} \\]</p>", "advice": "<p>For&nbsp; some quadratic expressions of the form \\[ ax^2+bx+c,\\] where $a \\neq 1$, we are able to factorise the expression into the form \\[(mx+p)(nx+q),\\]&nbsp;for integers $m$, $n$, $p$, and $q$.</p>\n<p>This is more difficult than when $a=1$, but there is a process which allows us to factorise certain quadratics of this type.&nbsp;</p>\n<p>The first step is to identify two numbers which multiply to give $ac$ and add to give $b$.</p>\n<p>So, for $\\simplify[unitFactor]{{a}x^2+{b}x+{c}}$, we have</p>\n<p>\\[ a \\times c = \\var{a*c} \\quad\\text{and}\\quad b=\\var{b}.\\]&nbsp;</p>\n<p>In this case,&nbsp;the numbers we need are&nbsp;<strong>$\\var{m*q}$</strong> and <strong>$\\var{n*p}$</strong>.</p>\n<p>Next, we want to rewrite our quadratic expression so that we have 2 linear terms with these coefficients:</p>\n<p>\\[\\simplify[unitFactor,!canonicalOrder]{{a}x^2+{m*q}x+{n*p}x+{c}}.\\]</p>\n<p>We are now able to factor the first two terms and the last two terms:</p>\n<p>\\[\\simplify[unitFactor,!canonicalOrder]{{m}x({n}x+{q})+{p}({n}x+{q})}.\\]</p>\n<p>We can see that we now have a common factor of $(\\simplify{{n}x+{q}})$, so rewriting our expression in terms of this factor, we get:</p>\n<p>\\[\\simplify[unitFactor]{({n}x+{q})({m}x+{p})},\\]</p>\n<p>as required.</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "m*n", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "m*q+n*p", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "{p*q}", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-7..7 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-7..7 except [0,p])", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..3 except n)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(m*n,p*q)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "m", "n", "p", "q"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({m}x+{p})*({n}x+{q})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "($n`?*x+`+-$n)($n`?*x+`+-$n)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}