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{"name": "HELM Book 1.3.5 exercises", "metadata": {"description": "<p>HELM Book 1.3.5 exercises</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "1.3.5.1a Factorise a monic quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Factorise a quadratic, x^2+bx+c. An exam can choose whether or not a=1. Part of HELM Book 1.3</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Practise these questions repeatedly until you can quickly, confidently and correctly factorise the expressions.</p>", "advice": "<p>$\\var{expr}=\\var{ans}$</p>\n<p>You can solve this by inspection or:&nbsp;</p>\n<p><span data-jme-visible=\"a=1\">Given a quadratic of the form $x^2+Bx+C$, find two numbers that multiply to give $C$ and add to give $B$</span></p>\n<p><span data-jme-visible=\"a=1\">$C=\\var{c}=(\\var{c1})\\times(\\var{c2})$ and $B=\\var{b}=(\\var{c1})+(\\var{c2})$</span></p>\n<p><span data-jme-visible=\"a=1\">so $\\var{expr}=\\var{ans}$</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\"><strong>Method 1 (the HELM book method)</strong></span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">\\[\\begin{align*}(\\var{a})(\\var{expr})&amp;=(\\var{a})(\\var{a}\\var{ve}^2)+(\\var{a})(\\var{b}\\var{ve})+(\\var{a})(\\var{c})\\\\ &amp;=(\\var{a}\\var{ve})^2+(\\var{b})(\\var{a}\\var{ve})+ \\var{a*c}\\end{align*}\\]</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">Let $\\var{a}\\var{ve}=u$, so</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">\\[\\begin{align*}(\\var{a})(\\var{expr}) &amp;= u^2+\\var{b}u+\\var{a*c}\\\\ &amp;= \\left(u+(\\var{c1*p2})\\right)\\left(u+(\\var{c2*p1})\\right)\\\\&amp;=(\\var{a}\\var{ve}+(\\var{c1*p2})(\\var{a}\\var{ve}+(\\var{c2*p1}))\\\\&amp;=(\\var{a})(\\var{ans})\\end{align*}\\]</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">so \\[\\var{expr}=\\var{ans}\\]</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\"><strong>Method 2 (the A-C method)</strong></span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">Given a quadratic of the form $Ax^2+Bx+C$, find two numbers that multiply to give $A\\times C$ and add to give $B$</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">$AC = (\\var{a})\\times(\\var{c}) = \\var{a*c} = (\\var{p1*c2})\\times(\\var{p2*c1})$ and $B = \\var{b}=(\\var{p1*c2})+(\\var{p2*c1})$</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">So</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">\\[\\begin{align*}\\var{expr}&amp;=\\var{a}\\var{ve}^2 + (\\var{p1*c2})\\var{ve} + (\\var{p2*c1})\\var{ve} + (\\var{c})\\\\&amp;=\\left[\\var{a}\\var{ve}^2 + (\\var{p1*c2})\\var{ve}\\right] + \\left[(\\var{p2*c1})\\var{ve} + (\\var{c})\\right]\\\\&amp;=(\\var{p1}\\var{ve})(\\var{term2})+(\\var{c1})(\\var{term2})\\\\&amp;=\\var{ans} \\end{align*}\\]</span></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "simplify(expression(a+\"*\"+v+\"^2+\"+b+\"*\"+v+\"+\"+c),[\"basic\",\"unitFactor\",\"zeroFactor\",\"zeroTerm\"])", "description": "", "templateType": "anything", "can_override": false}, "p1,p2": {"name": "p1,p2", "group": "Ungrouped variables", "definition": "[1,1]", "description": "<p>variable coefficients</p>", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a,b,c": {"name": "a,b,c", "group": "Ungrouped variables", "definition": "[p1*p2,c1*p2+c2*p1,c1*c2]", "description": "<p>(p1x+c1)(p2x+c2) = p1.p2x^2 + (c1.p2+c2.p1)x+c1.c2</p>", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "simplify(expression(\"(\"+p1+v+\"+\"+c1+\")(\"+p2+v+\"+\"+c2+\")\"),\"all\")", "description": "<p>(p1x+c1)(p2x+c2) = p1.p2x^2 + (c1.p2+c2.p1)x+c1.c2 = ax^2+bx+c</p>", "templateType": "anything", "can_override": false}, "ve": {"name": "ve", "group": "Ungrouped variables", "definition": "expression(v)", "description": "<p>for display in the advice</p>", "templateType": "anything", "can_override": false}, "term2": {"name": "term2", "group": "Ungrouped variables", "definition": "simplify(expression(\"(\"+p2+v+\"+\"+c2+\")\"),\"all\")", "description": "<p>for display in the advice</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p1,p2", "a,b,c", "v", "c1", "c2", "expr", "ans", "ve", "term2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "prompt": "<p>Factorise $\\var{expr}$.</p>", "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`+-$n`?*((`+- $v* ?`*) + `+- ?)*((`+- $v * ?`*) + `+-?) `| \n(`+-$n`?* $v * (`+- $v* ?`* + `+- ?))", "partialCredit": 0, "message": "You have not fully factorised the expression.", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.3.5.1 Factorise a quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Factorise a quadratic, ax^2+bx+c. An exam can choose whether or not a=1. Part of HELM Book 1.3</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Practise these questions repeatedly until you can quickly, confidently and correctly factorise the expressions.</p>", "advice": "<p>$\\var{expr}=\\var{ans}$</p>\n<p>You can solve this by inspection or:&nbsp;</p>\n<p><span data-jme-visible=\"a=1\">Given a quadratic of the form $x^2+Bx+C$, find two numbers that multiply to give $C$ and add to give $B$</span></p>\n<p><span data-jme-visible=\"a=1\">$C=\\var{c}=(\\var{c1})\\times(\\var{c2})$ and $B=\\var{b}=(\\var{c1})+(\\var{c2})$</span></p>\n<p><span data-jme-visible=\"a=1\">so $\\var{expr}=\\var{ans}$</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\"><strong>Method 1 (the HELM book method)</strong></span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">\\[\\begin{align*}(\\var{a})(\\var{expr})&amp;=(\\var{a})(\\var{a}\\var{ve}^2)+(\\var{a})(\\var{b}\\var{ve})+(\\var{a})(\\var{c})\\\\ &amp;=(\\var{a}\\var{ve})^2+(\\var{b})(\\var{a}\\var{ve})+ \\var{a*c}\\end{align*}\\]</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">Let $\\var{a}\\var{ve}=u$, so</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">\\[\\begin{align*}(\\var{a})(\\var{expr}) &amp;= u^2+\\var{b}u+\\var{a*c}\\\\ &amp;= \\left(u+(\\var{c1*p2})\\right)\\left(u+(\\var{c2*p1})\\right)\\\\&amp;=(\\var{a}\\var{ve}+(\\var{c1*p2})(\\var{a}\\var{ve}+(\\var{c2*p1}))\\\\&amp;=(\\var{a})(\\var{ans})\\end{align*}\\]</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">so \\[\\var{expr}=\\var{ans}\\]</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\"><strong>Method 2 (the A-C method)</strong></span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">Given a quadratic of the form $Ax^2+Bx+C$, find two numbers that multiply to give $A\\times C$ and add to give $B$</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">$AC = (\\var{a})\\times(\\var{c}) = \\var{a*c} = (\\var{p1*c2})\\times(\\var{p2*c1})$ and $B = \\var{b}=(\\var{p1*c2})+(\\var{p2*c1})$</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">So</span></p>\n<p><span data-jme-visible=\"a&lt;&gt;1\">\\[\\begin{align*}\\var{expr}&amp;=\\var{a}\\var{ve}^2 + (\\var{p1*c2})\\var{ve} + (\\var{p2*c1})\\var{ve} + (\\var{c})\\\\&amp;=\\left[\\var{a}\\var{ve}^2 + (\\var{p1*c2})\\var{ve}\\right] + \\left[(\\var{p2*c1})\\var{ve} + (\\var{c})\\right]\\\\&amp;=(\\var{p1}\\var{ve})(\\var{term2})+(\\var{c1})(\\var{term2})\\\\&amp;=\\var{ans} \\end{align*}\\]</span></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "simplify(expression(a+\"*\"+v+\"^2+\"+b+\"*\"+v+\"+\"+c),[\"basic\",\"unitFactor\",\"zeroFactor\",\"zeroTerm\"])", "description": "", "templateType": "anything", "can_override": false}, "p1,p2": {"name": "p1,p2", "group": "Ungrouped variables", "definition": "repeat(weighted_random([[1,0.7],[random(2,3,5),0.3]]),2)", "description": "<p>Factorised expression ax^2+bx+c is (p1.x-c1)(p1.x-c2). 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