// Numbas version: finer_feedback_settings
{"name": "Solving a perfect square by factorising", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "aa", "bb", "mid", "g", "gg", "c", "d"], "name": "Solving a perfect square by factorising", "tags": ["binomial", "factorisation", "Factorisation", "factorise", "perfect square", "quadratic", "quadratics", "solving", "square"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>Solve the following quadratic by factorisation:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td>$\\simplify{{aa}x^2+{mid}x+{bb}}$</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,&nbsp;there should only be one solution.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["xx", "x^2", "x**2", "x*x"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({gg})({a/g}x+{b/g})^2", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{-b}/{a}", "minValue": "{-b}/{a}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>Recall that $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2$ is called a perfect square.</p>\n<p>In fact, $\\simplify{{aa}x^2+{mid}x+{bb}}$ is also a perfect square, since $\\var{aa}x^2=(\\var{a}x)^2$,&nbsp;$\\var{bb}=\\simplify{({b})^2}$, and $\\var{mid}x=2(\\var{a}x)(\\var{b})$.</p>\n<p>That is,&nbsp;$\\simplify{{aa}x^2+{mid}x+{bb}}=\\simplify{({a}x+{b})^2}$.</p>\n<div data-jme-visible=\"g=1\">\n<p>Now, since&nbsp;$\\simplify{({a}x+{b})^2}=0$ we can take the (plus or minus) square root of both sides to get $\\simplify{({a}x+{b})}=0$. We can then solve this for $x$&nbsp;to find&nbsp;$x=\\simplify{-{b}/{a}}$.</p>\n<p></p>\n</div>\n<div data-jme-visible=\"g&gt;1\">\n<p style=\"text-align: left;\">Each bracket has a common factor of $\\var{g}$, so we can move both of them to the front, to&nbsp;write</p>\n<p style=\"text-align: center;\">$\\simplify{{aa}x^2+{mid}x+{bb}}=\\simplify{{gg}({a/g}x+{b/g})^2}$.</p>\n<p style=\"text-align: left;\">Now, using the null factor law, since&nbsp;$\\simplify{{gg}({a/g}x+{b/g})^2}=0$ we must have $\\simplify{{a/g}x+{b/g}}=0$.&nbsp;We can then solve this for $x$&nbsp;to find $x=\\simplify{{-b}/{a}}$.</p>\n</div>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>Solve the following quadratic by factorisation:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td>$\\simplify{{c[0]^2}/{c[1]^2} x^2+{2c[0]*d}/{(c[1]*c[2])}x+{d^2}/{c[2]^2}}$</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,&nbsp;there should only be one solution.</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["**2", "xx", "x^2", "(x)^2"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[0]}/{c[1]}x+{d}/{c[2]})^2", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["(", ")^2"], "partialCredit": 0}}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{-d*c[1]}/{c[0]*c[2]}", "minValue": "{-d*c[1]}/{c[0]*c[2]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>Recall that $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2$ is called a perfect square.</p>\n<p>In fact, $\\simplify{{c[0]^2}/{c[1]^2} x^2+{2c[0]*d}/{(c[1]*c[2])}x+{d^2}/{c[2]^2}}$&nbsp;is also a perfect square, since</p>\n<ul>\n<li>$\\simplify{{c[0]^2}/{c[1]^2}}x^2=\\left(\\simplify{{c[0]}/{c[1]}}x\\right)^2$,</li>\n<li>$\\simplify{{d^2}/{c[2]^2}}=\\left(\\simplify{{d}/{c[2]}}\\right)^2$ and</li>\n<li>$\\simplify{{2c[0]*d}/{(c[1]*c[2])}x}=2\\left(\\simplify{{c[0]}/{c[1]}x}\\right)\\left(\\simplify{{d}/{c[2]}}\\right)$.</li>\n</ul>\n<p>That is,&nbsp;</p>\n<p style=\"text-align: center;\">$\\simplify{{c[0]^2}/{c[1]^2} x^2+{2c[0]*d}/{(c[1]*c[2])}x+{d^2}/{c[2]^2}}=\\simplify{({c[0]}/{c[1]}x+{d}/{c[2]})^2}$.</p>\n<p></p>\n<p>Now, since&nbsp;$\\simplify{({c[0]}/{c[1]}x+{d}/{c[2]})^2}=0$ we can take the (plus or minus) square root of both sides to get $\\simplify{{c[0]}/{c[1]}x+{d}/{c[2]}}=0$. We can then solve this for $x$&nbsp;to find&nbsp;$x=\\simplify{{-d*c[1]}/{c[0]*c[2]}}$.</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": "127", "condition": ""}, "variables": {"a": {"definition": "random(2..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "aa": {"definition": "a^2", "templateType": "anything", "group": "Ungrouped variables", "name": "aa", "description": ""}, "c": {"definition": "shuffle(2..12)[0..3]", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-12..-2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "bb": {"definition": "b^2", "templateType": "anything", "group": "Ungrouped variables", "name": "bb", "description": ""}, "mid": {"definition": "2*a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "mid", "description": ""}, "gg": {"definition": "g^2", "templateType": "anything", "group": "Ungrouped variables", "name": "gg", "description": ""}, "g": {"definition": "gcd(a,b)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}