// Numbas version: finer_feedback_settings
{"feedback": {"allowrevealanswer": true, "intro": "", "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "showtotalmark": true, "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"name": "Factorising a monic quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["binomial", "factorisation", "Factorisation", "factorising", "factors", "Factors", "monic", "quadratic", "quadratics"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Factorise the following into linear factors. That is, write the quadratic as a product of terms that look like $ax+b$ where $a$ and $b$ are real numbers.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-12..12 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-12..12 except 0)", "description": "", "templateType": "anything", "can_override": false}, "const": {"name": "const", "group": "Ungrouped variables", "definition": "a*b", "description": "", "templateType": "anything", "can_override": false}, "linear": {"name": "linear", "group": "Ungrouped variables", "definition": "a+b", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "127"}, "ungrouped_variables": ["a", "b", "linear", "const"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{x^2+{linear}x+{const}}$ =&nbsp;[[0]].</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(x+{a})(x+{b})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "notallowed": {"strings": ["xx", "x^2", "x**2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Factorising a difference of two squares", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "ungrouped_variables": ["a", "b", "bb", "aa", "g", "gg", "c"], "tags": ["binomial", "difference of squares", "difference of two squares", "factorisation", "Factorisation", "factorise", "quadratic", "quadratics", "sum and minus"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "<p>$\\simplify{{aa}x^2-{bb}}$ =&nbsp;[[0]].</p>\n<p></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": "({gg})({a/g}x+{b/g})({a/g}x-{b/g})", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}], "steps": [{"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</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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "<p>$\\simplify{{c[0]}^2/{c[2]}^2 x^2-{c[1]}^2/{c[3]}^2}$ =&nbsp;[[0]].</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["**2", "xx", "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[2]}x+{c[1]}/{c[3]})({c[0]}/{c[2]}x-{c[1]}/{c[3]})", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Ensure you factorise the expression.</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}], "steps": [{"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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "<p>Factorise the following into linear factors. That is, write the quadratic as a product of terms that look like $ax+b$ where $a$ and $b$ are real numbers.</p>", "variable_groups": [], "variablesTest": {"maxRuns": "127", "condition": ""}, "variables": {"a": {"definition": "random(1..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..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "bb": {"definition": "b^2", "templateType": "anything", "group": "Ungrouped variables", "name": "bb", "description": ""}, "g": {"definition": "gcd(a,b)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "gg": {"definition": "g^2", "templateType": "anything", "group": "Ungrouped variables", "name": "gg", "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": []}]}, {"name": "Factorising a perfect square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["binomial", "factorisation", "Factorisation", "factorise", "perfect square", "quadratic", "quadratics", "square"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Factorise the following into&nbsp;squares, that is, $(ax+b)^2$ or if there is a common factor $k$, then&nbsp;$k(cx+d)^2$.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "aa": {"name": "aa", "group": "Ungrouped variables", "definition": "a^2", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "shuffle(2..12)[0..3]", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-12..12 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-12..-2)", "description": "", "templateType": "anything", "can_override": false}, "bb": {"name": "bb", "group": "Ungrouped variables", "definition": "b^2", "description": "", "templateType": "anything", "can_override": false}, "mid": {"name": "mid", "group": "Ungrouped variables", "definition": "2*a*b", "description": "", "templateType": "anything", "can_override": false}, "gg": {"name": "gg", "group": "Ungrouped variables", "definition": "g^2", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "gcd(a,b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "127"}, "ungrouped_variables": ["a", "b", "aa", "bb", "mid", "g", "gg", "c", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{{aa}x^2+{mid}x+{bb}}$ =&nbsp;[[0]].</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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</p>\n<ul>\n<li>$\\var{aa}x^2=(\\var{a}x)^2$,&nbsp;</li>\n<li>$\\var{bb}=\\simplify{({b})^2}$ and</li>\n<li>$\\var{mid}x=2(\\var{a}x)(\\var{b})$.</li>\n</ul>\n<p>That is,&nbsp;$\\simplify{{aa}x^2+{mid}x+{bb}}=\\simplify{({a}x+{b})^2}$.</p>\n<div data-jme-visible=\"g&gt;1\">\n<p>Each bracket has a common factor of $\\var{g}$, so we call move both of them to the front, to get</p>\n<p style=\"text-align: center;\">$\\simplify{{aa}x^2+{mid}x+{bb}}=\\simplify{{gg}({a/g}x+{b/g})^2}$.</p>\n</div>\n<p>You can always check your factorisation&nbsp;by expanding.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({gg})({a/g}x+{b/g})^2", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")^2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "notallowed": {"strings": ["xx", "x^2", "x**2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{{c[0]^2}/{c[1]^2} x^2+{2c[0]*d}/{(c[1]*c[2])}x+{d^2}/{c[2]^2}}$ =&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>You can always check your factorisation&nbsp;by expanding.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({c[0]}/{c[1]}x+{d}/{c[2]})^2", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")^2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "notallowed": {"strings": ["**2", "xx", "x^2", "(x)^2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Factorising a non-monic quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["binomial", "factorisation", "Factorisation", "factorise", "non-monic", "quadratic", "quadratics"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Factorise the following into linear factors. That is, write the quadratic as a product of terms that look like $ax+b$ where $a$ and $b$ are real numbers.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(2..6),2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(-6..6 except [-1,0,1]),2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "repeat(random(-6..6 except 0),2)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "repeat(random(-6..6 except 0),2)", "description": "", "templateType": "anything", "can_override": false}, "gcd_zero": {"name": "gcd_zero", "group": "Ungrouped variables", "definition": "gcd(c[0],d[0])", "description": "", "templateType": "anything", "can_override": false}, "gab_zero": {"name": "gab_zero", "group": "Ungrouped variables", "definition": "gcd(a[0],b[0])", "description": "", "templateType": "anything", "can_override": false}, "g_one": {"name": "g_one", "group": "Ungrouped variables", "definition": "gcd(c[1],d[1])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "127"}, "ungrouped_variables": ["a", "b", "c", "d", "g_one", "gab_zero", "gcd_zero"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$ =&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are a few different ways to do the working for these questions, here is one method that uses factorisation by grouping.</p>\n<p><br/>Given $\\simplify{{c[1]}x^2+{d[1]+b[1]*c[1]}x+{b[1]*d[1]}}$,&nbsp;we</p>\n<ol>\n<li style=\"text-align: left;\"><span data-jme-visible=\"1=1\">look for a common factor, in this case it is $\\var{g_one}$, and put it out the front:&nbsp;$\\simplify{{g_one}({c[1]/g_one}x^2+{(d[1]+b[1]*c[1])/g_one}x+{b[1]*d[1]/g_one})}$</span></li>\n<li><span data-jme-visible=\"1=1\">multiply the constant term and the coefficient of $x^2$ to get $\\var{c[1]/{g_one}*{b[1]*d[1]/g_one}}$</span></li>\n<li><span data-jme-visible=\"1=1\">find two numbers that multiply to give&nbsp;$\\var{c[1]/g_one*b[1]*d[1]/g_one}$ and add to give $\\var{(d[1]+b[1]*c[1])/g_one}$, in this case the numbers are $\\var{b[1]*c[1]/g_one}$ and $\\var{d[1]/g_one}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use these numbers to decompose the coefficient of the $x$ term, $\\simplify{{g_one}({c[1]/g_one}x^2+{b[1]*c[1]/g_one}x+{d[1]/g_one}x+{b[1]*d[1]/g_one})}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use factorisation by grouping to factorise the quadratic, that is:</span></li>\n<ul>\n<ul>\n<li><span data-jme-visible=\"1=1\">find the common factor for the first two terms, and then the second two terms, $\\simplify{{g_one}({c[1]/g_one}x(x+{b[1]}) + {d[1]/g_one}(x+{b[1]}) &nbsp;)}$</span></li>\n<li><span data-jme-visible=\"1=1\">realise $\\simplify{x+{b[1]}}$ is itself a common factor and take that out the front,&nbsp;$\\simplify{{g_one}(x+{b[1]})({c[1]/g_one}x + {d[1]/g_one} &nbsp;)}$</span></li>\n</ul>\n</ul>\n</ol>\n<p><span data-jme-visible=\"1=1\">As always, you can check your factorisation by expanding.</span></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{g_one}(x+{b[1]})({c[1]/g_one}*x+{d[1]/g_one})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, 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"<p>$\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$&nbsp;=&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are a few different ways to do the working for these questions, here is one method that uses factorisation by grouping.</p>\n<p><br/>Given $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+{b[0]*d[0]}}$,&nbsp;we</p>\n<ol>\n<li style=\"text-align: left;\"><span data-jme-visible=\"1=1\">look for a common factor, in this case it is $\\var{gab_zero*gcd_zero}$, and put it out the front: &nbsp;$\\simplify{{gab_zero*gcd_zero}({a[0]*c[0]/(gab_zero*gcd_zero)}x^2+{(a[0]*d[0]+b[0]*c[0])/(gab_zero*gcd_zero)}x+{b[0]*d[0]/(gab_zero*gcd_zero)})}$</span></li>\n<li><span data-jme-visible=\"1=1\">multiply the constant term and the coefficient of $x^2$ to get $\\var{a[0]*c[0]*b[0]*d[0]/(gab_zero*gcd_zero)^2}$</span></li>\n<li><span data-jme-visible=\"1=1\">find two numbers that multiply to give&nbsp;$\\var{a[0]*c[0]*b[0]*d[0]/(gab_zero*gcd_zero)^2}$ and add to give $\\var{(a[0]*d[0]+b[0]*c[0])/(gab_zero*gcd_zero)}$, in this case the numbers are $\\var{(a[0]*d[0])/(gab_zero*gcd_zero)}$ and $\\var{(b[0]*c[0])/(gab_zero*gcd_zero)}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use these numbers to decompose the coefficient of the $x$ term, $\\simplify{{gab_zero*gcd_zero}({a[0]*c[0]/(gab_zero*gcd_zero)}x^2+{(a[0]*d[0])/(gab_zero*gcd_zero)}x+{(b[0]*c[0])/(gab_zero*gcd_zero)}x+{b[0]*d[0]/(gab_zero*gcd_zero)})}$</span></li>\n<li><span data-jme-visible=\"1=1\">Use factorisation by grouping to factorise the quadratic, that is:</span></li>\n<ul>\n<li><span data-jme-visible=\"1=1\">find the common factor for the first two terms, and then the second two terms, $\\simplify{{gab_zero*gcd_zero}({a[0]/(gab_zero)}x({c[0]/gcd_zero}x+{d[0]/gcd_zero}) + {b[0]/(gab_zero)}({c[0]/gcd_zero}x+{d[0]/gcd_zero}) &nbsp;)}$</span></li>\n<li><span data-jme-visible=\"1=1\">realise $\\simplify{{c[0]/gcd_zero}x+{d[0]/gcd_zero}}$ is itself a common factor and take that out the front,&nbsp;$\\simplify{{gab_zero*gcd_zero}({c[0]/gcd_zero}x+{d[0]/gcd_zero})({a[0]/(gab_zero)}x+{b[0]/(gab_zero)}&nbsp; )}$</span></li>\n</ul>\n</ol>\n<p><span data-jme-visible=\"1=1\">As always, you can check your factorisation by expanding.</span></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{gab_zero*gcd_zero}({a[0]/gab_zero}*x+{b[0]/gab_zero})({c[0]/gcd_zero}*x+{d[0]/gcd_zero})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "notallowed": {"strings": ["xx", "x^2", "x**2"], "showStrings": false, "partialCredit": 0, "message": "<p>Ensure you factorise the expression.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "percentPass": "0", "navigation": {"reverse": true, "showresultspage": "oncompletion", "allowregen": true, "onleave": {"action": "none", "message": ""}, "preventleave": true, "showfrontpage": true, "browse": true}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Practice&nbsp;factorising different types of quadratics. Repeat as many times as you want.&nbsp;</p>"}, "timing": {"allowPause": true, "timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}}, "showQuestionGroupNames": false, "duration": 0, "name": "Factorising quadratics", "showstudentname": true, "type": "exam", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "extensions": [], "custom_part_types": [], "resources": []}