# What does completing the square mean?

Sep 16, 2014

Completing the square is a method that represents a quadratic equation as a combination of quadrilateral used to form a square.

The basis of this method is to discover a special value that when added to both sides of the quadratic that will create a perfect square trinomial.

That special value is found by evaluation the expression ${\left(\frac{b}{2}\right)}^{2}$ where $b$ is found in $a {x}^{2} + b x + c = 0$. Also in this explanation I assume that $a$ has a value of $1$.

$a {x}^{2} + b x + {\left(\frac{b}{2}\right)}^{2} = {\left(\frac{b}{2}\right)}^{2} - c$

${\left(a x + \frac{b}{2}\right)}^{2} = {\left(\frac{b}{2}\right)}^{2} - c$

$\sqrt{{\left(a x + \frac{b}{2}\right)}^{2}} = \pm \sqrt{{\left(\frac{b}{2}\right)}^{2} - c}$

$a x + \frac{b}{2} = \pm \sqrt{{\left(\frac{b}{2}\right)}^{2} - c}$

$a x = \pm \sqrt{{\left(\frac{b}{2}\right)}^{2} - c} - \frac{b}{2}$

A quadratic that is a perfect square is very easy to solve.

Please take a look at the video to see an example of completing the square visually.