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{"name": "Factorising Quadratics 1 - Basic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": [], "name": "Factorising Quadratics 1 - Basic", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p><strong>A worked example using the equation from Part a:</strong></p>\n<p style=\"text-align: center;\">$\\simplify{{f1}x^2+{f2}*x+{f3}}=0$</p>\n<p style=\"text-align: left;\">Firstly, start to find the possible common factors of the coefficient of $x$ (here, $\\var{f2}$), and the constant ($\\var{f3}$).</p>\n<p style=\"text-align: left;\">Here there are some possible factors of $\\var{f3}$, but we need to find two numbers which will sum to make $\\var{f2}$ (since the coefficient of $x^2$ is $1$ in this case).</p>\n<p style=\"text-align: left;\">We can use&nbsp;$\\var{c}$ and $\\var{d}$, since</p>\n<p style=\"text-align: left;\">$\\var{c}*\\var{d}=\\var{f3}$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$\\var{c}+\\var{d}=\\var{f2}$.</p>\n<p style=\"text-align: left;\">Once you have found the two numbers which will work, put them into two brackets like so:</p>\n<p style=\"text-align: center;\">$\\simplify{(x+{c})(x+{d})}$</p>\n<p style=\"text-align: left;\"><em><strong>This is your simplification.</strong></em></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">If you wanted to find the solutions for $x$, you would take each bracket and equate it to zero.</p>\n<p style=\"text-align: left;\">Some students may find this hard to see instantly, so the method is explained below.</p>\n<p style=\"text-align: left;\">Taking the first bracketed term,</p>\n<p style=\"text-align: left;\">($\\simplify{x+{c}}$)</p>\n<p style=\"text-align: left;\">and equating it to $0$</p>\n<p style=\"text-align: left;\">$\\simplify{x+{c}}=0$</p>\n<p style=\"text-align: left;\">then gives us a very simple equation.</p>\n<p style=\"text-align: left;\">From this, we can see&nbsp;$x=\\simplify{0-{c}}$.</p>\n<p style=\"text-align: left;\">The same goes for the second bracket:</p>\n<p style=\"text-align: left;\">$x=\\simplify{0-{d}}$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">A similar technique is used when the coefficient of $x^2$ is not $1$.</p>\n<p style=\"text-align: left;\">It simply must be taken into account that the factors will be multiplied by this.</p>\n<p style=\"text-align: left;\">For example, taking the equation $2x^2+7x+5$:</p>\n<p style=\"text-align: left;\">The only factors of $5$ are $5$ and $1$.</p>\n<p style=\"text-align: left;\">Creating the empty brackets,</p>\n<p style=\"text-align: left;\">$($$2x+$ &nbsp; &nbsp;$)($$x+$ &nbsp; &nbsp;$)$</p>\n<p style=\"text-align: left;\">we can see that they must be filled as follows:</p>\n<p style=\"text-align: left;\">$(2x+5)(x+1)$</p>\n<p style=\"text-align: left;\">as multiplying this out gives the original equation.</p>", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>$\\simplify{{f1}x^2+{f2}*x+{f3}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["^"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a}x+{c})({b}x+{d})", "marks": "1", "checkvariablenames": true, "checkingtype": "reldiff", "type": "jme"}], "steps": [{"prompt": "<p>Hint: try to find two numbers which will sum to the $x$ coefficient and multiply to the constant integer.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"prompt": "<p>$\\simplify{{f12}x^2+{f22}*x+{f32}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["^"], "partialCredit": "0"}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a2}x+{c2})({b2}x+{d2})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\simplify{{f13}x^2+{f23}*x+{f33}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["^"], "partialCredit": "0"}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a3}x+{c3})({b3}x+{d3})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p><strong>Simplify the following quadratics into two linear factors of the form $(ax+c)(bx+d)$.</strong></p>\n<p>Remember, the constants in your expression can be positive or negative.</p>", "variable_groups": [{"variables": ["a", "b", "c", "d", "f1", "f2", "f3"], "name": "Part a"}, {"variables": ["a2", "b2", "c2", "d2", "f12", "f22", "f32"], "name": "Part b"}, {"variables": ["a3", "b3", "c3", "d3", "f13", "f23", "f33"], "name": "Part c"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "1", "templateType": "anything", "group": "Part a", "name": "a", "description": ""}, "f1": {"definition": "a*b", "templateType": "anything", "group": "Part a", "name": "f1", "description": ""}, "c": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Part a", "name": "c", "description": ""}, "b": {"definition": "1", "templateType": "anything", "group": "Part a", "name": "b", "description": ""}, "f12": {"definition": "a2*b2", "templateType": "anything", "group": "Part b", "name": "f12", "description": ""}, "d": {"definition": "random(-5..5 except 0 except c)", "templateType": "anything", "group": "Part a", "name": "d", "description": ""}, "f32": {"definition": "c2*d2", "templateType": "anything", "group": "Part b", "name": "f32", "description": ""}, "f3": {"definition": "c*d", "templateType": "anything", "group": "Part a", "name": "f3", "description": ""}, "f2": {"definition": "(b*c)+(a*d)", "templateType": "anything", "group": "Part a", "name": "f2", "description": ""}, "f23": {"definition": "(b3*c3)+(a3*d3)", "templateType": "anything", "group": "Part c", "name": "f23", "description": ""}, "f22": {"definition": "(b2*c2)+(a2*d2)", "templateType": "anything", "group": "Part b", "name": "f22", "description": ""}, "a3": {"definition": "1", "templateType": "anything", "group": "Part c", "name": "a3", "description": ""}, "f33": {"definition": "c3*d3", "templateType": "anything", "group": "Part c", "name": "f33", "description": ""}, "b2": {"definition": "1", "templateType": "anything", "group": "Part b", "name": "b2", "description": ""}, "b3": {"definition": "1", "templateType": "anything", "group": "Part c", "name": "b3", "description": ""}, "c3": {"definition": "random(-15..15)", "templateType": "anything", "group": "Part c", "name": "c3", "description": ""}, "f13": {"definition": "a3*b3", "templateType": "anything", "group": "Part c", "name": "f13", "description": ""}, "a2": {"definition": "1", "templateType": "anything", "group": "Part b", "name": "a2", "description": ""}, "d2": {"definition": "random(-10..10 except c2)", "templateType": "anything", "group": "Part b", "name": "d2", "description": ""}, "d3": {"definition": "random(-15..15 except c3)", "templateType": "anything", "group": "Part c", "name": "d3", "description": ""}, "c2": {"definition": "random(-10..10)", "templateType": "anything", "group": "Part b", "name": "c2", "description": ""}}, "metadata": {"notes": "", "description": "<p>Factorising basic quadratics into linear expressions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}