// Numbas version: exam_results_page_options {"name": "Complete the square and find solutions [L4 Randomised]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "name": "Complete the square and find solutions [L4 Randomised]", "statement": "

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

The ^ symbol is required to write powers of algebraic symbols and * between integers and algebraic symbols, e.g. $5a^2$ should be written 5*a^2.

", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": ["l4", "L4"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solve a quadratic equation by completing the square. The roots are not pretty!

"}, "variable_groups": [], "rulesets": {}, "advice": "

Completing the square works by noticing that

\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]

So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.

a)

Rewrite $x^2+\\var{sml}x$ as $\\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$.

\\begin{align}
\\simplify[basic]{x^2+{sml}x+{big}} &= \\simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\\\
&= \\simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \\text{.}
\\end{align}

b)

We showed above that

\\[ \\simplify[basic]{x^2+{sml}x+{big}} = 0 \\]

is equivalent to

\\[ \\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \\text{.} \\]

We can then rearrange this equation to solve for $x$.

\\begin{align}
\\simplify{(x+{bits[0]})^2-{(bits[1])^2} } &= 0 \\\\
(x+\\var{bits[0]})^2 &= \\var{bits[1]^2} \\\\
x+\\var{bits[0]} &= \\pm \\var{bits[1]} \\\\
x &= -\\var{bits[0]} \\pm \\var{bits[1]} \\\\[2em]

x_1 &= \\var{-bits[0]-bits[1]} \\text{,}\\\\
x_2 &= \\var{-bits[0]+bits[1]} \\text{.}
\\end{align}

", "parts": [{"type": "gapfill", "gaps": [{"showFeedbackIcon": true, "type": "jme", "scripts": {}, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "notallowed": {"message": "

It doesn't look like you've completed the square.

", "showStrings": false, "partialCredit": 0, "strings": ["x^2"]}, "variableReplacements": [], "answer": "(x+{bits[0]})^2-{bits[1]^2}", "unitTests": [], "vsetRange": [0, 1], "musthave": {"message": "

It doesn't look like you've completed the square.

", "showStrings": false, "partialCredit": 0, "strings": [")^2"]}, "expectedVariableNames": [], "customMarkingAlgorithm": "", "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "showCorrectAnswer": true, "marks": 1}], "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "showFeedbackIcon": true, "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "prompt": "

Write the following expression in the form $a(x+b)^2-c$.

$\\simplify {x^2+{sml}x+{big}} = $ [[0]]

"}, {"type": "gapfill", "gaps": [{"showFeedbackIcon": true, "type": "jme", "scripts": {}, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "answer": "{-bits[0]-bits[1]}", "unitTests": [], "vsetRange": [0, 1], "expectedVariableNames": [], "customMarkingAlgorithm": "", "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "showCorrectAnswer": true, "marks": 1}, {"showFeedbackIcon": true, "type": "jme", "scripts": {}, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "answer": "{-bits[0]+bits[1]}", "unitTests": [], "vsetRange": [0, 1], "expectedVariableNames": [], "customMarkingAlgorithm": "", "checkVariableNames": false, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "showCorrectAnswer": true, "marks": 1}], "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "showFeedbackIcon": true, "sortAnswers": true, "customMarkingAlgorithm": "", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "prompt": "

Now solve the quadratic equation

\\[ \\simplify {x^2+{sml}x+{big}} = 0\\text{.} \\]

$x_1=$ [[0]]

$x_2=$ [[1]]

"}], "ungrouped_variables": ["big", "sml", "bits"], "variables": {"sml": {"templateType": "anything", "group": "Ungrouped variables", "name": "sml", "description": "

The coefficient of $x$ in the expanded quadratic.

", "definition": "2*bits[0]"}, "big": {"templateType": "anything", "group": "Ungrouped variables", "name": "big", "description": "

The constant term in the expanded quadratic.

", "definition": "bits[0]^2-bits[1]^2"}, "bits": {"templateType": "anything", "group": "Ungrouped variables", "name": "bits", "description": "

After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.

", "definition": "sort(shuffle(1..9)[0..2])"}}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}