Completing the square

Completing the square is a method used to rewrite a quadratic expression in a form that allows us to easily solve for $x$.

The goal is to express a quadratic equation in the form:

$$(x+p{)}^2=q$$

where $p$ and $q$ are constants.

Solve by completing the square

$$x^2+6x+5=0$$

Move the constant term to the other side.

$$x^2+6x=-5$$

Find ${\left({\displaystyle \frac{b}{2}}\right)}^2$

$${\left({\displaystyle \frac{6}{2}}\right)}^2=9$$

Add the 9 to both sides.

$$x^2+6x+9=-5+9$$

Rewrite the left side as a perfect square.

$$(x+3{)}^2=4$$

Solve for $x$.

$$x+3=\pm 2$$

$$x=-3\pm 2$$

So the solutions are:

$$x=-3+2=-1\text{and}x=-3-2=-5$$