// Numbas version: exam_results_page_options
{"name": "Musa's copy of 3 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": [{"name": "Musa's copy of 3 Factorising Quadratic Equations with $x^2$ Coefficients of 1", "tags": [], "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Factorise the following quadratic equations.</p>", "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>", "rulesets": {}, "extensions": [], "variables": {"v4": {"name": "v4", "group": "Part A ", "definition": "random(1..10 except -v3)", "description": "", "templateType": "anything"}, "v5": {"name": "v5", "group": "Part A ", "definition": "random(2..10)", "description": "", "templateType": "anything"}, "v6": {"name": "v6", "group": "Part A ", "definition": "-v5", "description": "", "templateType": "anything"}, "v2": {"name": "v2", "group": "Part A ", 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"random(4..8 except a2)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Part A ", "definition": "random(5..8)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Part A ", "definition": "random(5..8 except s1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part A ", "variables": ["v1", "v2", "v3", "v4", "v5", "v6", "v22", "v11", "a1", "a2", "a3", "a4", "s1", "s2"]}], "functions": {}, "preamble": {"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 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"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}]}], "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/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}