// Numbas version: exam_results_page_options
{"name": "Clare's copy of 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": [{"showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "steps": [{"marks": 0, "type": "information", "scripts": {}, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "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 style=\"line-height: 32px;\">Now, since $\\simplify{{g_one}(x+{b[1]})({c[1]/g_one}x + {d[1]/g_one} &nbsp;)}=0$, by the null factor law, either&nbsp;</span></p>\n<p style=\"text-align: center;\"><span style=\"line-height: 32px;\">$\\simplify{x+{b[1]}}=0$, or $\\simplify{{c[1]/g_one}x + {d[1]/g_one} &nbsp;=0}$.</span></p>\n<p><span style=\"line-height: 32px;\">Solving these equations results in&nbsp;</span></p>\n<p style=\"text-align: center;\"><span style=\"line-height: 32px;\">$x=\\var{-b[1]}$, or $x=\\simplify{{-d[1]}/{c[1]}}$.</span></p>"}], "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"marks": 1, "answersimplification": "all", "checkvariablenames": true, "variableReplacements": [], "musthave": {"strings": ["(", ")"], "message": "<p>Please factorise</p>", "showStrings": false, "partialCredit": 0}, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showpreview": true, "checkingtype": "absdiff", "expectedvariablenames": ["x"], "vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "jme", "notallowed": {"strings": ["^2", "**2", "(x)^2"], "message": "<p>Please factorise</p>", "showStrings": false, "partialCredit": 0}, "answer": "{g_one}(x+{b[1]})({c[1]/g_one}*x+{d[1]/g_one})", "vsetrangepoints": 5}, {"marks": 1, "answersimplification": "simplifyFractions", "checkvariablenames": false, "variableReplacements": [], "checkingaccuracy": 0.001, "showpreview": true, "checkingtype": "absdiff", "expectedvariablenames": [], "vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "jme", "answer": "set({-b[1]},{-d[1]}/{c[1]})", "showCorrectAnswer": true, "vsetrangepoints": 5}], "prompt": "<p>Solve the following quadratic by factorisation:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td>$\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">0</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\Longrightarrow$</td>\n<td style=\"text-align: right;\">&nbsp;[[0]]</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">0</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\Longrightarrow$</td>\n<td style=\"text-align: right;\">$x$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>Note:</strong> In the first gap, enter the quadratic in factored form.</p>\n<p><strong>Note:</strong> In the second gap, if $x=1,2$, enter <strong>set(1,2)</strong>.</p>"}, {"stepsPenalty": 0, "steps": [{"marks": 0, "type": "information", "scripts": {}, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "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 style=\"line-height: 32px;\">Now, since $\\simplify{{gab_zero*gcd_zero}({c[0]/gcd_zero}x+{d[0]/gcd_zero})({a[0]/(gab_zero)}x+{b[0]/(gab_zero)}&nbsp; )}=0$, by the null factor law, either&nbsp;</span></p>\n<p style=\"text-align: center;\"><span style=\"line-height: 32px;\">$\\simplify{{c[0]/gcd_zero}x+{d[0]/gcd_zero}}=0$, or $\\simplify{{a[0]/(gab_zero)}x+{b[0]/(gab_zero)} &nbsp;=0}$.</span></p>\n<p><span style=\"line-height: 32px;\">Solving these equations results in&nbsp;</span></p>\n<p style=\"text-align: center;\"><span style=\"line-height: 32px;\">$x=\\simplify{{-d[0]}/{c[0]}}$, or $x=\\simplify{{-b[0]}/{a[0]}}$.</span></p>"}], "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"marks": "2", "answersimplification": "all", "checkvariablenames": true, "variableReplacements": [], "musthave": {"strings": ["(", ")"], "message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "partialCredit": 0}, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showpreview": true, "checkingtype": "absdiff", "expectedvariablenames": ["x"], "vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "jme", "notallowed": {"strings": ["xx", "x^2", "x**2"], "message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "partialCredit": 0}, "answer": "{gab_zero*gcd_zero}({a[0]/gab_zero}*x+{b[0]/gab_zero})({c[0]/gcd_zero}*x+{d[0]/gcd_zero})", "vsetrangepoints": 5}, {"marks": 1, "answersimplification": "simplifyFractions", "checkvariablenames": false, "variableReplacements": [], "checkingaccuracy": 0.001, "showpreview": true, "checkingtype": "absdiff", "expectedvariablenames": [], "vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "jme", "answer": "set({-b[0]}/{a[0]},{-d[0]}/{c[0]})", "showCorrectAnswer": true, "vsetrangepoints": 5}], "prompt": "<p>Solve the following quadratic by factorisation:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td>$\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">0</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\Longrightarrow$</td>\n<td style=\"text-align: right;\">&nbsp;[[0]]</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">0</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\Longrightarrow$</td>\n<td style=\"text-align: right;\">$x$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>Note:</strong> In the first gap, enter the quadratic in factored form.</p>\n<p><strong>Note:</strong> In the second gap, if $x=1,2$, enter <strong>set(1,2)</strong>.</p>"}], "functions": {}, "variable_groups": [], "advice": "", "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "tags": ["binomial", "factorisation", "Factorisation", "factorise", "non-monic", "quadratic", "quadratics", "solving"], "type": "question", "variablesTest": {"maxRuns": "127", "condition": ""}, "preamble": {"css": "", "js": ""}, "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>", "name": "Clare's copy of Solving a non-monic quadratic by factorising", "rulesets": {}, "variables": {"b": {"group": "Ungrouped variables", "name": "b", "templateType": "anything", "definition": "repeat(random(-6..6 except 0),2)", "description": ""}, "c": {"group": "Ungrouped variables", "name": "c", "templateType": "anything", "definition": "repeat(random(-6..6 except [-1,0,1]),2)", "description": ""}, "gab_zero": {"group": "Ungrouped variables", "name": "gab_zero", "templateType": "anything", "definition": "gcd(a[0],b[0])", "description": ""}, "d": {"group": "Ungrouped variables", "name": "d", "templateType": "anything", "definition": "repeat(random(-6..6 except 0),2)", "description": ""}, "a": {"group": "Ungrouped variables", "name": "a", "templateType": "anything", "definition": "repeat(random(2..6),2)", "description": ""}, "g_one": {"group": "Ungrouped variables", "name": "g_one", "templateType": "anything", "definition": "gcd(c[1],d[1])", "description": ""}, "gcd_zero": {"group": "Ungrouped variables", "name": "gcd_zero", "templateType": "anything", "definition": "gcd(c[0],d[0])", "description": ""}}, "ungrouped_variables": ["a", "b", "c", "d", "g_one", "gab_zero", "gcd_zero"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "notes": "<p>I could use !noLeadingMinus in simplify to avoid it rearranging</p>", "description": ""}, "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}