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.

i

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 $(x+\color{red}{\frac{p}{2}})^2\color{red}{-(\frac{p}{2})^2}$
i

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


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