// Numbas version: exam_results_page_options
{"name": "Factorising a non-monic quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Factorising a non-monic quadratic", "tags": ["binomial", "factorisation", "Factorisation", "factorise", "non-monic", "quadratic", "quadratics"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Factorise the following into linear factors. That is, write the quadratic as a product of terms that look like $ax+b$ where $a$ and $b$ are real numbers.</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(2..6),2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(-6..6 except [-1,0,1]),2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "repeat(random(-6..6 except 0),2)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "repeat(random(-6..6 except 0),2)", "description": "", "templateType": "anything", "can_override": false}, "gcd_zero": {"name": "gcd_zero", "group": "Ungrouped variables", "definition": "gcd(c[0],d[0])", "description": "", "templateType": "anything", "can_override": false}, "gab_zero": {"name": "gab_zero", "group": "Ungrouped variables", "definition": "gcd(a[0],b[0])", "description": "", "templateType": "anything", "can_override": false}, "g_one": {"name": "g_one", "group": "Ungrouped variables", "definition": "gcd(c[1],d[1])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "127"}, "ungrouped_variables": ["a", "b", "c", "d", "g_one", "gab_zero", "gcd_zero"], "variable_groups": [], "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, "exploreObjective": null, "prompt": "<p>$\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$ =&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are a few different ways to do the working for these questions, here is one method that uses factorisation by grouping.</p>\n<p><br/>Given $\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$,&nbsp;we</p>\n<ol>\n<li style=\"text-align: left;\"><span data-jme-visible=\"1=1\">look for a common factor, in this case it is $\\var{g_one}$, and put it out the front:&nbsp;$\\simplify{{g_one}({c[1]/g_one}x^2+{(d[1]+b[1]*c[1])/g_one}x+{b[1]*d[1]/g_one})}$</span></li>\n<li><span data-jme-visible=\"1=1\">multiply the constant term and the coefficient of $x^2$ to get $\\var{c[1]/{g_one}*{b[1]*d[1]/g_one}}$</span></li>\n<li><span data-jme-visible=\"1=1\">find two numbers that multiply to give&nbsp;$\\var{c[1]/g_one*b[1]*d[1]/g_one}$ and add to give $\\var{(d[1]+b[1]*c[1])/g_one}$, in this case the numbers are $\\var{b[1]*c[1]/g_one}$ and $\\var{d[1]/g_one}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use these numbers to decompose the coefficient of the $x$ term, $\\simplify{{g_one}({c[1]/g_one}x^2+{b[1]*c[1]/g_one}x+{d[1]/g_one}x+{b[1]*d[1]/g_one})}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use factorisation by grouping to factorise the quadratic, that is:</span></li>\n<ul>\n<ul>\n<li><span data-jme-visible=\"1=1\">find the common factor for the first two terms, and then the second two terms, $\\simplify{{g_one}({c[1]/g_one}x(x+{b[1]}) + {d[1]/g_one}(x+{b[1]}) &nbsp;)}$</span></li>\n<li><span data-jme-visible=\"1=1\">realise $\\simplify{x+{b[1]}}$ is itself a common factor and take that out the front,&nbsp;$\\simplify{{g_one}(x+{b[1]})({c[1]/g_one}x + {d[1]/g_one} &nbsp;)}$</span></li>\n</ul>\n</ul>\n</ol>\n<p><span data-jme-visible=\"1=1\">As always, you can check your factorisation by expanding.</span></p>"}], "gaps": [{"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": "{g_one}(x+{b[1]})({c[1]/g_one}*x+{d[1]/g_one})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Please factorise</p>"}, "notallowed": {"strings": ["^2", "**2", "(x)^2"], "showStrings": false, "partialCredit": 0, "message": "<p>Please factorise</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$&nbsp;=&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are a few different ways to do the working for these questions, here is one method that uses factorisation by grouping.</p>\n<p><br/>Given $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$,&nbsp;we</p>\n<ol>\n<li style=\"text-align: left;\"><span data-jme-visible=\"1=1\">look for a common factor, in this case it is $\\var{gab_zero*gcd_zero}$, and put it out the front: &nbsp;$\\simplify{{gab_zero*gcd_zero}({a[0]*c[0]/(gab_zero*gcd_zero)}x^2+{(a[0]*d[0]+b[0]*c[0])/(gab_zero*gcd_zero)}x+{b[0]*d[0]/(gab_zero*gcd_zero)})}$</span></li>\n<li><span data-jme-visible=\"1=1\">multiply the constant term and the coefficient of $x^2$ to get $\\var{a[0]*c[0]*b[0]*d[0]/(gab_zero*gcd_zero)^2}$</span></li>\n<li><span data-jme-visible=\"1=1\">find two numbers that multiply to give&nbsp;$\\var{a[0]*c[0]*b[0]*d[0]/(gab_zero*gcd_zero)^2}$ and add to give $\\var{(a[0]*d[0]+b[0]*c[0])/(gab_zero*gcd_zero)}$, in this case the numbers are $\\var{(a[0]*d[0])/(gab_zero*gcd_zero)}$ and $\\var{(b[0]*c[0])/(gab_zero*gcd_zero)}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use these numbers to decompose the coefficient of the $x$ term, $\\simplify{{gab_zero*gcd_zero}({a[0]*c[0]/(gab_zero*gcd_zero)}x^2+{(a[0]*d[0])/(gab_zero*gcd_zero)}x+{(b[0]*c[0])/(gab_zero*gcd_zero)}x+{b[0]*d[0]/(gab_zero*gcd_zero)})}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use factorisation by grouping to factorise the quadratic, that is:</span></li>\n<ul>\n<li><span data-jme-visible=\"1=1\">find the common factor for the first two terms, and then the second two terms, $\\simplify{{gab_zero*gcd_zero}({a[0]/(gab_zero)}x({c[0]/gcd_zero}x+{d[0]/gcd_zero}) + {b[0]/(gab_zero)}({c[0]/gcd_zero}x+{d[0]/gcd_zero}) &nbsp;)}$</span></li>\n<li><span data-jme-visible=\"1=1\">realise $\\simplify{{c[0]/gcd_zero}x+{d[0]/gcd_zero}}$ is itself a common factor and take that out the front,&nbsp;$\\simplify{{gab_zero*gcd_zero}({c[0]/gcd_zero}x+{d[0]/gcd_zero})({a[0]/(gab_zero)}x+{b[0]/(gab_zero)}&nbsp; )}$</span></li>\n</ul>\n</ol>\n<p><span data-jme-visible=\"1=1\">As always, you can check your factorisation by expanding.</span></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{gab_zero*gcd_zero}({a[0]/gab_zero}*x+{b[0]/gab_zero})({c[0]/gcd_zero}*x+{d[0]/gcd_zero})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "notallowed": {"strings": ["xx", "x^2", "x**2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}