Forget the Quadratic Formula… Complete the Square

An alternative and more applicable approach

Photo by Annie Spratt on Unsplash

The first real formula that I remember learning was none other than the quadratic formula. Even visually, it just looked like the epitome of what a complex mathematical equation should look like: cool arithmetic combined with fancy square root signs and variables. After the initial learning process of applying it to basic equations, the quadratic formula never left my side during my time in middle school. However, I did notice that the work proceeded to become more and more difficult, especially when the problems involved funky numbers. My teachers began to feed me new, interesting methods of solving the same type of equation in a more efficient manner. Enter Completing the Square.

The idea of completing the square is simple: we want to obtain a perfect square, so let’s just change whatever we have to get what we want. To take a step back, please refer to the examples below of what a perfect square is and isn’t.

• x² + 4x + 4 = (x + 2)² ==> PERFECT SQUARE
• x² + 10x + 25 = (x + 5)² ==> PERFECT SQUARE
• x² + 14x + 50 ==> NOT A PERFECT SQUARE

We want to obtain this form because we can simply take the square root of both sides of the equation and solve for the x variable. Compared to using the quadratic formula, this approach is much cleaner.

Examples

Now, let us look at an example.

• x² + 8x + 12 = 0

Here we have an equation where it is possible to manually crank out the quadratic formula. However, completing the square may prove to be a much simpler alternative. Please view the general steps below for solving equations using the completing the square method.

Steps

(1) Manipulate the side of the equation containing the x’s into the form of a perfect square by adding or subtracting a constant value. Remember to do the same to the other side of the equation as well.

(2) In the most basic form, take the positive and negative square root of both sides of the equation. If you’re dealing with higher degrees, take whatever root necessary.