// Numbas version: exam_results_page_options {"name": "Complete the square and find solutions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"bits": {"templateType": "anything", "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])", "name": "bits", "group": "Ungrouped variables"}, "big": {"templateType": "anything", "description": "

The constant term in the expanded quadratic.

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

The coefficient of $x$ in the expanded quadratic.

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

\n

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

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

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

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Now solve the quadratic equation

\n

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

\n

$x_1=$ [[0]]

\n

or

\n

$x_2=$ [[1]]

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Completing the square works by noticing that

\n

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

\n

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

\n

a)

\n

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

\n

\\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}

\n

b)

\n

We showed above that

\n

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

\n

is equivalent to

\n

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

\n

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

\n

\\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}

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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\".

\n

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

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

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