# Completing the square

Quite often, it is not immediately obvious how to factor the quadratic equation. For example, it is not easy at all to see how to factor x^2+3x-5=0 , since the solutions are x=\frac{-3+\sqrt{29}}{2} and x=\frac{-3-\sqrt{29}}{2}.

For these cases the method of completing the square can be used. This method works in every case. It makes use of the algebraic identity of the perfect square x^2 +2kx+k^2 = (x+k)^2 .

For example, x^2 +4x+4 = (x+2)^2 and x^2 -10x+25 = (x-5)^2 .

Note that in these examples the constant term is the square of **half **the coefficient of the x term. Completing the square literally means to add a number so that the quadratic expression can be written as a perfect square.

Take a look at the following graphical representation of completing the square. A factor of (\frac{b}{2})^2 is added to literally complete the square.

If that’s not clear yet, maybe this nice animation will clear things up.

Let’s take a look at the quadratic equation x^2+3x+1=0 . Consider the terms with an x only. In this case x^2+3x . What number do we need to add to turn the expression into a perfect square? The square of half the coefficient of the x term, hence (\frac{3}{2})^2=\frac{9}{4} .

To solve a quadratic equation the following steps should be taken:

- Move the constant to the other side
- Add a factor of (\frac{b}{2})^2 to both sides of the equation
- Write the perfect square, so that x appears only once in the complete equation
- Take the square root of both sides
- Write down the solution(s)

**Example**

Let’s try it on the quadratic equation x^2+3x+1=0 .

Bring the constant to the other side

$$ x^2+3x=-1 $$ Add a factor of (\frac{b}{2})^2 to both sides of the equation

$$ x^2+3x+\left(\frac{3}{2}\right)^2=-1+\left(\frac{3}{2}\right)^2 $$ Simplify

$$ x^2+3x+\frac{9}{4}=-1+\frac{9}{4}=\frac{5}{4} $$ Write the perfect square, so that x appears only once in the complete equation

$$ \left(x+\frac{3}{2}\right)^2=\frac{5}{4} $$ Take the square root of both sides

$$ x+\frac{3}{2}=\pm \sqrt{\frac{5}{4}} $$ Write down the solutions

$$ x=-\frac{3}{2} + \sqrt{\frac{5}{4}}~\text{or}~ x=-\frac{3}{2} – \sqrt{\frac{5}{4}} $$

A great advantage of the method of completing the square is that, unlike factoring, it works for irrational and complex roots as well.

**Example**

Let’s solve the quadratic equation 2x^2+4x+4=0

First divide everything by 2

$$ x^2 +2x+2=0 $$ Bring the constant to the other side

$$ x^2 +2x=-2 $$ Add a factor of (\frac{b}{2})^2 to both sides of the equation

$$ x^2+2x +1 =-2+1 $$ Write the perfect square

$$ (x+1)^2=-1 $$ Take the square root of both sides

$$ x+1 =\pm \sqrt{-1} $$ Bring the constant to the right

$$ x= -1\pm \sqrt{-1} $$ Finally simplify the root by using the identity $i^2=-1$

$$ x=-1\pm i $$