// Numbas version: exam_results_page_options {"name": "Solving a non-monic quadratic by factorising", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "g_one", "gab_zero", "gcd_zero"], "name": "Solving a non-monic quadratic by factorising", "tags": ["binomial", "factorisation", "Factorisation", "factorise", "non-monic", "quadratic", "quadratics", "solving"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "
Solve the following quadratic by factorisation:
\n\n | $\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$ | \n$=$ | \n0 | \n
$\\Longrightarrow$ | \n[[0]] | \n$=$ | \n0 | \n
$\\Longrightarrow$ | \n$x$ | \n$=$ | \n[[1]] | \n
Note: In the first gap, enter the quadratic in factored form.
\nNote: In the second gap, if $x=1,2$, enter set(1,2).
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "Please factorise
", "showStrings": false, "strings": ["^2", "**2", "(x)^2"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{g_one}(x+{b[1]})({c[1]/g_one}*x+{d[1]/g_one})", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "Please factorise
", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "simplifyFractions", "scripts": {}, "answer": "set({-b[1]},{-d[1]}/{c[1]})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "There are a few different ways to do the working for these questions, here is one method that uses factorisation by grouping.
\n
Given $\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$, we
Now, since $\\simplify{{g_one}(x+{b[1]})({c[1]/g_one}x + {d[1]/g_one} )}=0$, by the null factor law, either
\n$\\simplify{x+{b[1]}}=0$, or $\\simplify{{c[1]/g_one}x + {d[1]/g_one} =0}$.
\nSolving these equations results in
\n$x=\\var{-b[1]}$, or $x=\\simplify{{-d[1]}/{c[1]}}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "Solve the following quadratic by factorisation:
\n\n | $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$ | \n$=$ | \n0 | \n
$\\Longrightarrow$ | \n[[0]] | \n$=$ | \n0 | \n
$\\Longrightarrow$ | \n$x$ | \n$=$ | \n[[1]] | \n
Note: In the first gap, enter the quadratic in factored form.
\nNote: In the second gap, if $x=1,2$, enter set(1,2).
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "Ensure you factorise the expression.
", "showStrings": false, "strings": ["xx", "x^2", "x**2"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{gab_zero*gcd_zero}({a[0]/gab_zero}*x+{b[0]/gab_zero})({c[0]/gcd_zero}*x+{d[0]/gcd_zero})", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "Ensure you factorise the expression.
", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "simplifyFractions", "scripts": {}, "answer": "set({-b[0]}/{a[0]},{-d[0]}/{c[0]})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "There are a few different ways to do the working for these questions, here is one method that uses factorisation by grouping.
\n
Given $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$, we
Now, since $\\simplify{{gab_zero*gcd_zero}({c[0]/gcd_zero}x+{d[0]/gcd_zero})({a[0]/(gab_zero)}x+{b[0]/(gab_zero)} )}=0$, by the null factor law, either
\n$\\simplify{{c[0]/gcd_zero}x+{d[0]/gcd_zero}}=0$, or $\\simplify{{a[0]/(gab_zero)}x+{b[0]/(gab_zero)} =0}$.
\nSolving these equations results in
\n$x=\\simplify{{-d[0]}/{c[0]}}$, or $x=\\simplify{{-b[0]}/{a[0]}}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "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.
", "variable_groups": [], "variablesTest": {"maxRuns": "127", "condition": ""}, "variables": {"a": {"definition": "repeat(random(2..6),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "repeat(random(-6..6 except [-1,0,1]),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "repeat(random(-6..6 except 0),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "repeat(random(-6..6 except 0),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "gcd_zero": {"definition": "gcd(c[0],d[0])", "templateType": "anything", "group": "Ungrouped variables", "name": "gcd_zero", "description": ""}, "gab_zero": {"definition": "gcd(a[0],b[0])", "templateType": "anything", "group": "Ungrouped variables", "name": "gab_zero", "description": ""}, "g_one": {"definition": "gcd(c[1],d[1])", "templateType": "anything", "group": "Ungrouped variables", "name": "g_one", "description": ""}}, "metadata": {"notes": "I could use !noLeadingMinus in simplify to avoid it rearranging
", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "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/"}]}