// Numbas version: exam_results_page_options {"name": "Factorising a difference of two squares", "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", "bb", "aa", "g", "gg", "c"], "name": "Factorising a difference of two squares", "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": "

$\\simplify{{aa}x^2-{bb}}$ = [[0]].

\n

\n

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

Ensure you factorise the expression.

", "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": "

Ensure you factorise the expression.

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}], "steps": [{"prompt": "
\n

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. 

\n

Recalling that $(a+b)(a-b)=a^2-b^2$, we have

\n

$\\simplify{{aa}x^2-{bb}}=(\\simplify{{a}x+{b}})(\\simplify{{a}x-{b}})$ 

\n
\n

Notice there is a common factor of $\\var{gg}$ that we can deal with first

\n

$\\simplify{{aa}x^2-{bb}}=\\simplify{{gg}({aa/gg}x^2-{bb/gg})}.$

\n

Next, notice the remaining expression is a difference of two squares. Recalling that $(a+b)(a-b)=a^2-b^2$, we have

\n

$\\simplify{{aa}x^2-{bb}}=\\simplify{{gg}({aa/gg}x^2-{bb/gg})}=\\simplify{({gg})({a/g}x+{b/g})({a/g}x-{b/g})}$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "

$\\simplify{{c[0]}^2/{c[2]}^2 x^2-{c[1]}^2/{c[3]}^2}$ = [[0]].

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

Ensure you factorise the expression.

", "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": "

Ensure you factorise the expression.

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}], "steps": [{"prompt": "

We should recognise this as a difference of two squares, where $\\simplify{{c[0]}^2/{c[2]}^2 x^2}$ 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

\n

$\\simplify{{c[0]}^2/{c[2]}^2 x^2-{c[1]}^2/{c[3]}^2}=\\simplify{({c[0]}/{c[2]}x+{c[1]}/{c[3]})({c[0]}/{c[2]}x-{c[1]}/{c[3]})}.$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "

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.

", "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": []}], "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/"}]}