Forget the Quadratic Formula… Complete the Square

An alternative and more applicable approach

Albert Ming


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