// Numbas version: exam_results_page_options
{"name": "Terry's copy of Solving a difference of two squares by factorising", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "name": "Terry's copy of Solving a difference of two squares by factorising", "parts": [{"marks": 0, "stepsPenalty": "1", "prompt": "<p>Solve the following quadratic by factorisation:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td>$\\simplify{{aa}x^2-{bb}}$</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, if $x=1,2$, enter <strong>set(1,2)</strong>.</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "gaps": [{"answer": "({gg})({a/g}x+{b/g})({a/g}x-{b/g})", "marks": 1, "answersimplification": "all", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkvariablenames": true, "scripts": {}, "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "variableReplacements": [], "showpreview": true, "musthave": {"strings": ["(", ")"], "message": "<p>Ensure you factorise the expression.</p>", "partialCredit": 0, "showStrings": false}, "vsetrange": [0, 1], "expectedvariablenames": ["x"], "notallowed": {"strings": ["xx", "x^2", "x**2"], "message": "<p>Ensure you factorise the expression.</p>", "partialCredit": 0, "showStrings": false}, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "answersimplification": "all", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff", "scripts": {}, "showpreview": true, "type": "jme", "vsetrange": [0, 1], "answer": "set({b}/{a},-{b}/{a})", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"marks": 0, "showCorrectAnswer": true, "scripts": {}, "prompt": "<div data-jme-visible=\"g=1\">\n<p>Since $\\simplify{{aa}x^2}$ is $\\simplify{{a}x}$ squared and $\\var{bb}$ is $\\var{b}$ squared, we can recognise $\\simplify{{aa}x^2-{bb}}$ as a difference of two squares.&nbsp;</p>\n<p>Recalling that $(a+b)(a-b)=a^2-b^2$, we&nbsp;have</p>\n<p style=\"text-align: center;\">$\\simplify{{aa}x^2-{bb}}=(\\simplify{{a}x+{b}})(\\simplify{{a}x-{b}})$&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">Now, by the null factor law, either</p>\n<p style=\"text-align: center;\">$\\simplify{{a}x+{b}}=0$ or&nbsp;$\\simplify{{a}x-{b}}=0$.</p>\n<p style=\"text-align: left;\">Solving these equations results in</p>\n<p style=\"text-align: center;\">$x=\\simplify{-{b}/{a}}$ or $x=\\simplify{{b}/{a}}$.</p>\n</div>\n<p><span style=\"line-height: 32px;\" data-jme-visible=\"g&gt;1\">Notice there is a common factor of $\\var{gg}$ that we can deal with first</span></p>\n<p style=\"text-align: center;\"><span style=\"line-height: 32px;\" data-jme-visible=\"g&gt;1\">$\\simplify{{aa}x^2-{bb}}=\\simplify{{gg}({aa/gg}x^2-{bb/gg})}.$</span></p>\n<p style=\"text-align: left;\"><span style=\"line-height: 32px;\" data-jme-visible=\"g&gt;1\">Next, notice the remaining expression is a difference of two squares.&nbsp;Recalling that $(a+b)(a-b)=a^2-b^2$, we&nbsp;have</span></p>\n<p style=\"text-align: center;\"><span style=\"line-height: 32px;\" data-jme-visible=\"g&gt;1\">$\\simplify{{aa}x^2-{bb}}=\\simplify{{gg}({aa/gg}x^2-{bb/gg})}=\\simplify{({gg})({a/g}x+{b/g})({a/g}x-{b/g})}$</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"g&gt;1\">Now, by the null factor law, either</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"g&gt;1\">$\\simplify{{a/g}x+{b/g}}=0$ or&nbsp;$\\simplify{{a/g}x-{b/g}}=0$.</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"g&gt;1\">Solving these equations results in</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"g&gt;1\">$x=\\simplify{-{b}/{a}}$ or $x=\\simplify{{b}/{a}}$.</span></p>", "type": "information", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}]}, {"marks": 0, "stepsPenalty": "1", "prompt": "<p>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;\">$\\simplify{{c[0]}^2/{c[2]}^2 x^2-{c[1]}^2/{c[3]}^2}$</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></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, if $x=1,2$, enter <strong>set(1,2)</strong>.</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "gaps": [{"answer": "({c[0]}/{c[2]}x+{c[1]}/{c[3]})({c[0]}/{c[2]}x-{c[1]}/{c[3]})", "marks": "2", "answersimplification": "all", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkvariablenames": true, "scripts": {}, "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "variableReplacements": [], "showpreview": true, "musthave": {"strings": ["(", ")"], "message": "<p>Ensure you factorise the expression.</p>", "partialCredit": 0, "showStrings": false}, "vsetrange": [0, 1], "expectedvariablenames": ["x"], "notallowed": {"strings": ["**2", "xx", "x^2"], "message": "<p>Ensure you factorise the expression.</p>", "partialCredit": 0, "showStrings": false}, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "answersimplification": "all", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff", "scripts": {}, "showpreview": true, "type": "jme", "vsetrange": [0, 1], "answer": "set({num}/{den}, {-num}/{den})", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"marks": 0, "showCorrectAnswer": true, "scripts": {}, "prompt": "<p>We should recognise this as a difference of two squares, where&nbsp;$\\simplify{{c[0]}^2/{c[2]}^2 x^2}$ is $\\left(\\simplify{{c[0]}/{c[2]}x}\\right)^2$ and $\\simplify{{c[1]}^2/{c[3]}^2}$ is $\\left(\\simplify{{c[1]}/{c[3]}}\\right)^2$. Therefore</p>\n<p style=\"text-align: center;\">$\\simplify{{c[0]}^2/{c[2]}^2 x^2-{c[1]}^2/{c[3]}^2}=\\simplify{({c[0]}/{c[2]}x+{c[1]}/{c[3]})({c[0]}/{c[2]}x-{c[1]}/{c[3]})}.$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">Now, by the null factor law, either</p>\n<p style=\"text-align: center;\">$\\simplify{{c[0]}/{c[2]}x+{c[1]}/{c[3]}}=0$ or&nbsp;$\\simplify{{c[0]}/{c[2]}x-{c[1]}/{c[3]}}=0$.</p>\n<p style=\"text-align: left;\">Solving these equations results in</p>\n<p style=\"text-align: center;\">$x=\\simplify{-{c[1]*c[2]}/{c[3]*c[0]}}$ or $x=\\simplify{{c[1]*c[2]}/{c[3]*c[0]}}$.</p>", "type": "information", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}]}], "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "statement": "", "tags": ["binomial", "difference of squares", "difference of two squares", "factorisation", "Factorisation", "factorise", "quadratic", "quadratics", "solving", "sum and minus"], "ungrouped_variables": ["a", "b", "bb", "aa", "g", "gg", "c", "num", "den", "solb"], "variablesTest": {"maxRuns": "127", "condition": ""}, "advice": "", "functions": {}, "showQuestionGroupNames": false, "variables": {"g": {"name": "g", "description": "", "templateType": "anything", "definition": "gcd(a,b)", "group": "Ungrouped variables"}, "gg": {"name": "gg", "description": "", "templateType": "anything", "definition": "g^2", "group": "Ungrouped variables"}, "bb": {"name": "bb", "description": "", "templateType": "anything", "definition": "b^2", "group": "Ungrouped variables"}, "num": {"name": "num", "description": "", "templateType": "anything", "definition": "c[1]*c[2]", "group": "Ungrouped variables"}, "b": {"name": "b", "description": "", "templateType": "anything", "definition": "random(1..12)", "group": "Ungrouped variables"}, "den": {"name": "den", "description": "", "templateType": "anything", "definition": "c[3]*c[0]", "group": "Ungrouped variables"}, "c": {"name": "c", "description": "", "templateType": "anything", "definition": "shuffle(2..12)[0..4]", "group": "Ungrouped variables"}, "a": {"name": "a", "description": "", "templateType": "anything", "definition": "random(1..12)", "group": "Ungrouped variables"}, "aa": {"name": "aa", "description": "", "templateType": "anything", "definition": "a^2", "group": "Ungrouped variables"}, "solb": {"name": "solb", "description": "", "templateType": "anything", "definition": "set(num/den, -num/den)", "group": "Ungrouped variables"}}, "rulesets": {}, "type": "question", "metadata": {"description": "", "notes": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}]}