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{"name": "MATH6058 Factorising Quadratic Equations with $x^2$ Coefficients of 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "advice": "<p>Quadratic equations&nbsp;of the form</p>\n<p>\\[x^2+bx+c=0\\]</p>\n<p>can be factorised to&nbsp;create&nbsp;an equation of the form</p>\n<p>\\[(x+m)(x+n)=0\\text{.}\\]</p>\n<p>When we expand a factorised quadratic expression we obtain</p>\n<p>\\[(x+m)(x+n)=x^2+(m+n)x+(m \\times n)\\text{.}\\]</p>\n<p>To factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.</p>\n<h4>a)</h4>\n<p>\\[\\simplify{x^2+{v1+v2}x+{v1*v2}=0}\\]</p>\n<p>We need to find two values that add together to make $\\var{v1+v2}$ and multiply together to make $\\var{v1*v2}$.</p>\n<p>\\[\\begin{align}<br/>\\var{v1} \\times \\var{v2}&amp;=\\var{v1*v2}\\\\<br/>\\var{v1}+\\var{v2}&amp;=\\var{v1+v2}\\\\<br/>\\end{align} \\]</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v1})(x+{v2})}=0\\text{.}\\]</p>\n<p></p>\n<h4>b)</h4>\n<p>We can begin factorising by finding factors of $\\var{v3*v4}$ that add together to give $\\var{v3+v4}$.</p>\n<p>\\[\\begin{align}<br/>\\var{v3} \\times \\var{v4}&amp;=\\var{v3*v4}\\\\<br/>\\var{v3}+\\var{v4}&amp;=\\var{v3+v4}\\\\<br/>\\end{align} \\]</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v3})(x+{v4})}=0\\text{.}\\]</p>\n<h4>c)</h4>\n<p>When factorising the quadratic expression</p>\n<p>\\[\\simplify{x^2+{v5*v6}=0}\\]</p>\n<p>we need to find two values that add together to make $0$ and multiply together to make $\\var{v5*v6}$.</p>\n<p>\\begin{align}<br/>\\var{v5} \\times \\var{v6}&amp; = \\var{v5*v6}\\\\<br/>\\simplify[]{ {v5} + {v6}} &amp;= 0 \\\\<br/>\\end{align}</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v5})(x+{v6})}=0\\text{.}\\]</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Factorise the following quadratic equations.</p>", "extensions": [], "preamble": {"css": "", "js": "question.is_factorised = function(part,penalty) {\n  penalty = penalty || 0;\n  if(part.credit>0) {\n    // Parse the student's answer as a syntax tree\n    var studentTree = Numbas.jme.compile(part.studentAnswer,Numbas.jme.builtinScope);\n\n    // Create the pattern to match against \n    // we just want two sets of brackets, each containing two terms\n    // or one of the brackets might not have a constant term\n    // or for repeated roots, you might write (x+a)^2\n    var rule = Numbas.jme.compile('m_all(m_any(x,x+m_pm(m_number),x^m_number,(x+m_pm(m_number))^m_number))*m_nothing');\n\n    // Check the student's answer matches the pattern. \n    var m = 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"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\simplify{x^2+{v1+v2}x+{v1*v2}=0}$</p>\n<p>[[0]]&nbsp;$=0$</p>", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "useCustomName": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true}, {"gaps": [{"vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "unitTests": [], "valuegenerators": [{"name": "x", "value": ""}], "answer": "(x+{v3})(x+{v4})", "customMarkingAlgorithm": "", "scripts": {}, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true, "mustmatchpattern": {"pattern": "((`+-x + `+-$n`?)^$n`?)`* * $z", "nameToCompare": "", "partialCredit": 0, "message": "Your answer is not fully factorised."}, "customName": "", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "marks": 1, "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "type": "jme", "checkVariableNames": false}], "sortAnswers": false, "customName": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\simplify{x^2+{v3+v4}x+{v3*v4}}=0$</p>\n<p>[[0]] $=0$</p>\n<p><em></em></p>", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "useCustomName": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true}, {"gaps": [{"vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "unitTests": [], "valuegenerators": [{"name": "x", "value": ""}], "answer": "(x+{v5})(x+{v6})", "customMarkingAlgorithm": "", "scripts": {}, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true, "mustmatchpattern": {"pattern": "((`+-x + `+-$n`?)^$n`?)`* * $z", "nameToCompare": "", "partialCredit": 0, "message": "Your answer is not fully factorised."}, "customName": "", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "marks": 1, "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "type": "jme", "checkVariableNames": false}], "sortAnswers": false, "customName": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p><em></em>$\\simplify{x^2+{v5*v6}}=0$</p>\n<p>[[0]] $=0$</p>", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "useCustomName": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true}], "functions": {}, "name": "MATH6058 Factorising Quadratic Equations with $x^2$ Coefficients of 1", "variables": {"v4": {"name": "v4", "definition": "random(1..10 except -v3)", "description": "", "templateType": "anything", "group": "Part A "}, "v5": {"name": "v5", "definition": "random(2..10)", "description": "", "templateType": "anything", "group": "Part A "}, "v6": {"name": "v6", "definition": "-v5", "description": "", "templateType": "anything", "group": "Part A "}, "v2": {"name": "v2", "definition": "random(2..6 except v1)", "description": "", "templateType": "anything", "group": "Part A "}, "v3": {"name": "v3", "definition": "random(-8..-1)", "description": "", "templateType": "anything", "group": "Part A "}, "v1": {"name": "v1", "definition": "random(1..10)", "description": "", "templateType": "anything", "group": "Part A "}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Factorise three quadratic equations of the form $x^2+bx+c$.</p>\n<p>The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.</p>"}, "ungrouped_variables": [], "variable_groups": [{"variables": ["v1", "v2", "v3", "v4", "v5", "v6"], "name": "Part A "}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}