Completing the Square

Completing the square involves manipulating an expression so that it becomes a perfect square and we can factor it.

For example, if we have $x^2 + 4x + 2 = 0$ , we can make the expression on the left a perfect square by adding $2$ to both sides ( because $(x+2)^2 = x^2 + 4x + 4$ ). This makes it significantly easier to solve the equation.

$\begin{aligned} x^2 + 4x + 2 &= 0 \\ x^2 + 4x + 4 &= 2 \\ (x+2)^2 &= 2 \\ x &= -2 \pm \sqrt{2} \end{aligned}$

More generally,

$ax^2 + bx + c = a ( x + \frac{b}{2a} )^2 + \frac{ 4ac - b^2 } { 4a}.$

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$