// Numbas version: exam_results_page_options {"name": "Completing the square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)-a^2", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1.0..4.5#0.5)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "s1", "b"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Completing the square", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "(x+{a})^2+{b}", "musthave": {"message": "

please input in the form \$(x+a)^2+b\$

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", "showStrings": false, "partialCredit": 0, "strings": ["x^2", "x*x", "x x", "x(", "x*("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

\$\\simplify{x^2+{2*a}x+ {a^2+b}} = \\phantom{{}}\$ [[0]].

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Given the quadratic \$q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}\$ we complete the square by:

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1. Halving the coefficient of \$x\$ gives \$\\var{a}\$

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2. Work out \$\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}\$.
This gives the first two terms of \$q(x)\$.

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3. But the constant term \$\\simplify[all]{{a^2}}\$ in \$p(x)\$ is not the same as in \$q(x)\$ – so we need to adjust by adding on a suitable constant to \$p(x)\$.

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

Put the following quadratic expression in the form \$(x+a)^2+b\$ for suitable numbers \$a\$ and \$b\$.

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Note that you have to input your answer in the form \$(x+a)^2+b\$  and  the numbers \$a,\\;b\$ must be input exactly.

", "tags": ["MAS1601", "Steps", "algebra", "algebraic manipulation", "checked2015", "complete the square", "completing the square", "mas1601", "quadratics", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

14/7/2015

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5/08/2012:

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Checked calculation.OK.

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Find \$c\$ and \$d\$ such that \$x^2+ax+b = (x+c)^2+d\$.

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Given the quadratic \$q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}\$ we complete the square by:

\n

1. Halving the coefficient of \$x\$ gives \$\\var{a}\$

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2. Work out \$\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}\$.
This gives the first two terms of \$q(x)\$.

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3. But the constant term \$\\simplify[all]{{a^2}}\$ in \$p(x)\$ is not the same as in \$q(x)\$, so we need to adjust by adding on \$\\simplify[std,!fractionNumbers]{{a^2+b}-{a^2}={b}}\$ to \$p(x)\$.
Hence we get \\[q(x) = \\simplify[all]{p(x)+{b} = (x+{a})^2+{b}}=\\simplify[all]{ (x+{a})^2+{b}}\\]

", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}