// Numbas version: exam_results_page_options
{"name": "Factorising Quadratics 2 - Intermediate", "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 2 - Intermediate", "tags": [], "advice": "<p><strong>Repeated roots questions, 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;\">These are solved in the exact same way as the previous set of quadratics, but the 'roots' (solutions) are&nbsp;the same.</p>\n<p style=\"text-align: left;\">This type of equation is described as having 'repeated roots'.</p>\n<p style=\"text-align: left;\">The graphed functions would only touch the $x$-axis once either at their peak or trough.</p>\n<p style=\"text-align: left;\">Looking at the above equation, you may be able to instantly see that the brackets must both include the quantity $\\var{c}$.</p>\n<p style=\"text-align: left;\">If not, don't worry, just go through the same process of trial and error as taught previously until you reach your solution of</p>\n<p style=\"text-align: left;\">$\\simplify{({a}x+{c})({b}x+{d})}$.</p>\n<p style=\"text-align: left;\">Then, noticing this can be further simplified, write the final simplification as:</p>\n<p style=\"text-align: center;\">$\\simplify{({a}x+{c})^2}$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"><em></em></p>\n<p style=\"text-align: left;\"><em></em></p>\n<p style=\"text-align: left;\"><em>Note:</em>&nbsp;If the $x^2$ term has a coefficient and simplifies to, for example, $(2x+4)(x+4)$, this is not defined as having repeated roots, since the solutions are:</p>\n<p style=\"text-align: left;\">$2x+4=0$&emsp;&emsp;so&emsp;&emsp;&emsp;$x=\\frac{-4}{2}=-2$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$x+4=0$&emsp;&emsp;&nbsp;&nbsp;so&emsp;&emsp;&emsp;$x=-4$</p>\n<p style=\"text-align: left;\">so are in fact different.</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;\"><strong>Zero constants equations, using Part d:</strong></p>\n<p style=\"text-align: center;\">$\\simplify{{f14}x^2+{f24}*x+{f34}}=0$</p>\n<p style=\"text-align: left;\">Some equations will not have a constant at the end, meaning one of the constants is zero.</p>\n<p style=\"text-align: left;\">In this case, if there is still an $x$ term, you know one of the constants must still be non-zero.</p>\n<p style=\"text-align: left;\">You must therefore, take out a factor of a number $\\times x$.</p>\n<p style=\"text-align: left;\">Here, the factor could be $\\simplify{{a4}x}$ or&nbsp;$\\simplify{{b4}x}$.</p>\n<p style=\"text-align: left;\">It doesn't matter which you choose, but in this example we will use&nbsp;$\\simplify{{a4}x}$.</p>\n<p style=\"text-align: left;\">In order to pull out this factor, the rest of the equation must be divided by&nbsp;$\\simplify{{a4}x}$:</p>\n<p style=\"text-align: left;\">$\\frac{\\simplify{{f14}x^2+{f24}*x+{f34}}}{\\simplify{{a4}*x}}=\\simplify{{f14}x/{a4}+{f24}/{a4}}$.</p>\n<p style=\"text-align: left;\">Therefore, the solution is:</p>\n<p style=\"text-align: center;\">$\\simplify{{a4}x*({f14}x/{a4}+{f24}/{a4})}$.</p>", "rulesets": {}, "parts": [{"prompt": "<p>$\\simplify{{f1}x^2+{f2}*x+{f3}}$</p>\n<p>$=$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a}x+{c})({b}x+{d})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1], "musthave": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["(", ")"], "partialCredit": 0}}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "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": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a2}x+{c2})({b2}x+{d2})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "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": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a3}x+{c3})({b3}x+{d3})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "<p>$\\simplify{{f14}x^2+{f24}*x+{f34}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a4}x+{c4})({b4}x+{d4})", "marks": "3", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "steps": [{"prompt": "<p>Hint:&nbsp;Remember one or more of the constants could be zero.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "<p><strong>Simplify the following quadratics into two linear factors in the form $(ax+c)(bx+d)$.</strong></p>\n<p>Quadratics are defined as having 'repeated roots' if the both roots in the solution are the same. This can be written as $(x+a)(x+a)$, but can be simplified to $(x+a)^2$.</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"}, {"variables": ["a4", "b4", "c4", "d4", "f14", "f24", "f34"], "name": "Part d"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"f1": {"definition": "a*b", "templateType": "anything", "group": "Part a", "name": "f1", "description": ""}, "f2": {"definition": "(b*c)+(a*d)", "templateType": "anything", "group": "Part a", "name": "f2", "description": ""}, "f3": {"definition": "c*d", "templateType": "anything", "group": "Part a", "name": "f3", "description": ""}, "b4": {"definition": "random(1..2 except a4)", "templateType": "anything", "group": "Part d", "name": "b4", "description": ""}, "b2": {"definition": "random(1..3)", "templateType": "anything", "group": "Part b", "name": "b2", "description": ""}, "b3": {"definition": "random(2..3)", "templateType": "anything", "group": "Part c", "name": "b3", "description": ""}, "d4": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part d", "name": "d4", "description": ""}, "d2": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part b", "name": "d2", "description": ""}, "d3": {"definition": "random(-15..15 except 0)", "templateType": "anything", "group": "Part c", "name": "d3", "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": ""}, "f24": {"definition": "(b4*c4)+(a4*d4)", "templateType": "anything", "group": "Part d", "name": "f24", "description": ""}, "a3": {"definition": "random(1..3)", "templateType": "anything", "group": "Part c", "name": "a3", "description": ""}, "a2": {"definition": "random(2..3)", "templateType": "anything", "group": "Part b", "name": "a2", "description": ""}, "a4": {"definition": "random(1..3)", "templateType": "anything", "group": "Part d", "name": "a4", "description": ""}, "c3": {"definition": "random(-10..10)", "templateType": "anything", "group": "Part c", "name": "c3", "description": ""}, "c2": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part b", "name": "c2", "description": ""}, "c4": {"definition": "0", "templateType": "anything", "group": "Part d", "name": "c4", "description": ""}, "a": {"definition": "1", "templateType": "anything", "group": "Part a", "name": "a", "description": ""}, "c": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part a", "name": "c", "description": ""}, "b": {"definition": "1", "templateType": "anything", "group": "Part a", "name": "b", "description": ""}, "d": {"definition": "c", "templateType": "anything", "group": "Part a", "name": "d", "description": ""}, "f32": {"definition": "c2*d2", "templateType": "anything", "group": "Part b", "name": "f32", "description": ""}, "f33": {"definition": "c3*d3", "templateType": "anything", "group": "Part c", "name": "f33", "description": ""}, "f34": {"definition": "c4*d4", "templateType": "anything", "group": "Part d", "name": "f34", "description": ""}, "f12": {"definition": "a2*b2", "templateType": "anything", "group": "Part b", "name": "f12", "description": ""}, "f13": {"definition": "a3*b3", "templateType": "anything", "group": "Part c", "name": "f13", "description": ""}, "f14": {"definition": "a4*b4", "templateType": "anything", "group": "Part d", "name": "f14", "description": ""}}, "metadata": {"notes": "", "description": "<p>Factorising further 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/"}]}