Write the expression $ax^2+bx+c$ in completed square form $a(x+p)^2+k$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Rewrite the expression \\[\\simplify{{a}x^2+{b}x+{c}}\\] in completed square form $a(x+k)^2+p$ for suitable numbers $a, k$ and $p$.
\n\n
Your answer must be input exactly in this form.
", "advice": "(a)
\nGiven the quadratic $q(x)=\\simplify{{a}x^2+{b}x+ {c}}$, one way of completing the square is to simply expand the term $a(x+k)^2+p$ and equate coefficients to the quadratic term.
\nExpanding $a(x+k)^2+p$ gives $ax^2 + 2akx + ak^2 + p$.
\nEquating coefficients gives:
\n$x^2:\\quad \\var{a}=a$ or $a=\\var{a}$.
\n$x:\\quad \\var{b}= 2ak \\implies k = \\simplify{{b}/(2{a})}$
\nconstant: $\\var{c}=ak^2+p \\implies p=\\var{c}-ak^2=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.
\nNow, put these together to obtain: $\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.
\n(b)
\nCompleting the square for this function gives:$\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.
\nThe function attains its turning point when $\\quad \\simplify{{a}(x+{b}/(2{a}))^2}=0$
\nSo the $x$ coordinate of the turning point is $\\simplify{-{b}/(2{a})}$.
\nThe $y$ coordinate is $k=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.
\nHence the turning point is $\\quad (\\simplify{-{b}/(2{a})},\\simplify[all, fractionNumbers]{{c-b^2/(4a)}})$
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", "stepsPenalty": "2", "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": "Given the quadratic $q(x)=\\simplify{{a}x^2+{b}x+ {c}}$, one way of completing the square is to simply expand the term $a(x+k)^2+p$ and equate coefficients to the quadratic term.
\nExpanding $a(x+k)^2+p$ gives $ax^2 + 2akx + ak^2 + p$.
\nEquating coefficients gives:
\n$x^2:\\quad \\var{a}=a$ or $a=\\var{a}$.
\n$x:\\quad \\var{b}= 2ak \\implies k = \\simplify{{b}/(2{a})}$
\nconstant: $\\var{c}=ak^2+p \\implies p=\\var{c}-ak^2=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.
\nNow, put these together to obtain: $\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.
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"}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": "Input your answer in the form $(x+a)^2+b$.
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\nTurning point:$\\quad \\phantom{}$[[0]].
", "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": "Completing the square for this function gives:$\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.
\nThe function attains its turning point when $\\quad \\simplify{{a}(x+{b}/(2{a}))^2}=0$
\nSo the $x$ coordinate of the turning point is $\\simplify{-{b}/(2{a})}$.
\nThe $y$ coordinate is $k=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.
\nHence the turning point is $\\quad (\\simplify{-{b}/(2{a})},\\simplify[all, fractionNumbers]{{c-b^2/(4a)}})$
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