// Numbas version: exam_results_page_options
{"name": "Solving a monic quadratic 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", "linear", "const", "monicsoln"], "name": "Solving a monic quadratic by factorising", "tags": ["binomial", "factorisation", "Factorisation", "factorising", "factors", "Factors", "monic", "quadratic", "quadratics", "solving"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "<p>&nbsp;Solve the following quadratic by factorisation:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: right;\">$\\simplify{x^2+{linear}x+{const}}$</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><strong>Note:</strong> In the first gap, enter the quadratic in factored form.</p>\n<p><strong>Note:</strong> In the second gap, if $x=1,2$, enter <strong>set(1,2)</strong>.</p>\n<p></p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["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": "(x+{a})(x+{b})", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{monicsoln}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "<p>Since $(x+a)(x+b)=x^2+(a+b)x+ab$, when we are factorising a quadratic, such as $x^2+cx+d$, we must find the numbers $a$ and $b$ such that $c=a+b$ and $d=ab$.</p>\n<p></p>\n<p>In the case of&nbsp;$\\simplify{x^2+{linear}x+{const}}$ we ask</p>\n<p style=\"text-align: center;\"><em>what two numbers add to give $\\var{linear}$ and multiply to give $\\var{const}$?&nbsp;</em></p>\n<p style=\"text-align: left;\">Therefore the numbers must be $\\var{a}$ and $\\var{b}$, that is&nbsp;</p>\n<p style=\"text-align: center;\">$\\simplify{x^2+{linear}x+{const}}=(\\simplify{x+{a}})(\\simplify{x+{b}}).$</p>\n<p style=\"text-align: left;\">You can check this by expanding the binomial product.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}, {"prompt": "<p>Now, using the null factor law we have, either $\\simplify{x+{a}}=0$ or $\\simplify{x+{b}}=0$. In otherwords, either $x=\\var{-a}$ or $\\var{-b}$.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td style=\"text-align: right;\">$\\simplify{x^2+{linear}x+{const}}$</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;\">$ (\\simplify{x+{a}})(\\simplify{x+{b}})$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">0</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\Longrightarrow$</td>\n<td style=\"text-align: right;\">$x$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{-a},\\var{-b}$</td>\n</tr>\n</tbody>\n</table>\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(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "const": {"definition": "a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "const", "description": ""}, "linear": {"definition": "a+b", "templateType": "anything", "group": "Ungrouped variables", "name": "linear", "description": ""}, "monicsoln": {"definition": "set(-a,-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "monicsoln", "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/"}]}