Math Quadratic equations Completing the square

# Completing the square

Completing the square is used to transform a term so that one can apply the first or second binomial formula backwards.

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### Method

1. Square term with the form:
$x^2+px$
Important: If a coefficient is before $x^2$, it must be excluded beforehand

2. Completing the square
$x^2+px\color{red}{+(\frac{p}{2})^2-(\frac{p}{2})^2}$

3. Apply binomial formula backwards $(x+\color{red}{\frac{p}{2}})^2\color{red}{-(\frac{p}{2})^2}$
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### Hint

Completing the square is often used to get a quadratic function in the vertex shape or to solve a mixed quadratic equation.

### Example

Put the function $f(x)=2x^2-80x$ into the vertex shape

1. #### Exclude coefficient before $x^2$

$f(x)=2x^2-80x$
$f(x)=2(x^2-40x)$
2. #### Apply completing the square

$f(x)=2(x^2-40x+\color{red}{(\frac{40}{2})^2}-\color{red}{(\frac{40}{2})^2})$
3. #### Apply 2nd binomial formula backwards and resolve parenthesis

$f(x)=2(x^2-40x+\color{red}{20^2}-\color{red}{20^2})$
$f(x)=2((x-\color{red}{20})^2-\color{red}{400})$
$f(x)=2(x-20)^2-800$