// Numbas version: exam_results_page_options {"name": "MATH6058 Factorising Quadratic Equations with $x^2$ Coefficients of 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "advice": "

\n

\$x^2+bx+c=0\$

\n

can be factorised to create an equation of the form

\n

\$(x+m)(x+n)=0\\text{.}\$

\n

When we expand a factorised quadratic expression we obtain

\n

\$(x+m)(x+n)=x^2+(m+n)x+(m \\times n)\\text{.}\$

\n

To factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.

\n

#### a)

\n

\$\\simplify{x^2+{v1+v2}x+{v1*v2}=0}\$

\n

We need to find two values that add together to make $\\var{v1+v2}$ and multiply together to make $\\var{v1*v2}$.

\n

\\\begin{align} \\var{v1} \\times \\var{v2}&=\\var{v1*v2}\\\\ \\var{v1}+\\var{v2}&=\\var{v1+v2}\\\\ \\end{align} \

\n

So the factorised form of the equation is

\n

\$\\simplify{(x+{v1})(x+{v2})}=0\\text{.}\$

\n

\n

#### b)

\n

We can begin factorising by finding factors of $\\var{v3*v4}$ that add together to give $\\var{v3+v4}$.

\n

\\\begin{align} \\var{v3} \\times \\var{v4}&=\\var{v3*v4}\\\\ \\var{v3}+\\var{v4}&=\\var{v3+v4}\\\\ \\end{align} \

\n

So the factorised form of the equation is

\n

\$\\simplify{(x+{v3})(x+{v4})}=0\\text{.}\$

\n

#### c)

\n

\n

\$\\simplify{x^2+{v5*v6}=0}\$

\n

we need to find two values that add together to make $0$ and multiply together to make $\\var{v5*v6}$.

\n

\\begin{align}
\\var{v5} \\times \\var{v6}& = \\var{v5*v6}\\\\
\\simplify[]{ {v5} + {v6}} &= 0 \\\\
\\end{align}

\n

So the factorised form of the equation is

\n

\$\\simplify{(x+{v5})(x+{v6})}=0\\text{.}\$

", "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

", "extensions": [], "preamble": {"css": "", "js": "question.is_factorised = function(part,penalty) {\n penalty = penalty || 0;\n if(part.credit>0) {\n // Parse the student's answer as a syntax tree\n var studentTree = Numbas.jme.compile(part.studentAnswer,Numbas.jme.builtinScope);\n\n // Create the pattern to match against \n // we just want two sets of brackets, each containing two terms\n // or one of the brackets might not have a constant term\n // or for repeated roots, you might write (x+a)^2\n var rule = Numbas.jme.compile('m_all(m_any(x,x+m_pm(m_number),x^m_number,(x+m_pm(m_number))^m_number))*m_nothing');\n\n // Check the student's answer matches the pattern. \n var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n // If not, take away marks\n if(!m) {\n part.multCredit(penalty,'Your answer is not fully factorised.');\n }\n }\n}"}, "tags": [], "parts": [{"gaps": [{"vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "unitTests": [], "valuegenerators": [{"name": "x", "value": ""}], "answer": "(x+{v1})(x+{v2})", "customMarkingAlgorithm": "", "scripts": {}, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true, "mustmatchpattern": {"pattern": "((+-x + +-$n?)^$n?)* * $z", "nameToCompare": "", "partialCredit": 0, "message": "Your answer is not fully factorised."}, "customName": "", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "marks": 1, "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "type": "jme", "checkVariableNames": false}], "sortAnswers": false, "customName": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "$\\simplify{x^2+{v1+v2}x+{v1*v2}=0}$\n [[0]]$=0$", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "useCustomName": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true}, {"gaps": [{"vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "unitTests": [], "valuegenerators": [{"name": "x", "value": ""}], "answer": "(x+{v3})(x+{v4})", "customMarkingAlgorithm": "", "scripts": {}, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true, "mustmatchpattern": {"pattern": "((+-x + +-$n?)^$n?)* *$z", "nameToCompare": "", "partialCredit": 0, "message": "Your answer is not fully factorised."}, "customName": "", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "marks": 1, "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "type": "jme", "checkVariableNames": false}], "sortAnswers": false, "customName": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "

$\\simplify{x^2+{v3+v4}x+{v3*v4}}=0$

\n

[[0]] $=0$

\n

", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "useCustomName": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true}, {"gaps": [{"vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "unitTests": [], "valuegenerators": [{"name": "x", "value": ""}], "answer": "(x+{v5})(x+{v6})", "customMarkingAlgorithm": "", "scripts": {}, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true, "mustmatchpattern": {"pattern": "((+-x + +-$n?)^$n?)`* * $z", "nameToCompare": "", "partialCredit": 0, "message": "Your answer is not fully factorised."}, "customName": "", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "marks": 1, "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "type": "jme", "checkVariableNames": false}], "sortAnswers": false, "customName": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "$\\simplify{x^2+{v5*v6}}=0$\n [[0]]$=0$", "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "useCustomName": false, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "showFeedbackIcon": true}], "functions": {}, "name": "MATH6058 Factorising Quadratic Equations with$x^2$Coefficients of 1", "variables": {"v4": {"name": "v4", "definition": "random(1..10 except -v3)", "description": "", "templateType": "anything", "group": "Part A "}, "v5": {"name": "v5", "definition": "random(2..10)", "description": "", "templateType": "anything", "group": "Part A "}, "v6": {"name": "v6", "definition": "-v5", "description": "", "templateType": "anything", "group": "Part A "}, "v2": {"name": "v2", "definition": "random(2..6 except v1)", "description": "", "templateType": "anything", "group": "Part A "}, "v3": {"name": "v3", "definition": "random(-8..-1)", "description": "", "templateType": "anything", "group": "Part A "}, "v1": {"name": "v1", "definition": "random(1..10)", "description": "", "templateType": "anything", "group": "Part A "}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Factorise three quadratic equations of the form$x^2+bx+c\$.

\n

The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.

"}, "ungrouped_variables": [], "variable_groups": [{"variables": ["v1", "v2", "v3", "v4", "v5", "v6"], "name": "Part A "}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}