Forget the Quadratic Formula… Complete the Square
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.
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.
(1) Manipulate the side of the equation containing 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.
(3) Solve for x in both of the equations generated from taking the positive and negative square roots.
Referring back to the original example, follow the first step by adding 4 to both sides of the equation.
x² + 8x + 16 = 4
(x + 4)² = 4
Then take the positive and negative square root
x + 4 = 2
x + 4 = -2
Finally, solve for x
x = -2 and -6
Applications in Calculus
Truth be told, completing the square sort of vanished from my everyday use after middle school. Maybe it was because we eventually had to move on from quadratics. However, the method made its long-awaited appearance in calculus class, much to my surprise. It turns out that completing the square can help us solve for indefinite integrals! Refer to the example problem below.
Here is the plan for tackling this complex problem.
(1) Manipulate and complete the square for the expression inside of the denominator.
(2) Factor out 1/3² from the denominator of the integral and rewrite the term involving x.
(3) Now, the integral should remind us of the integral of arctan(x). Finish off the task by completing the proper u-substitution.
In hindsight, the idea behind completing the square is one that makes a lot of sense. We are turning what would have been long and arduous calculations into clean, succinct expressions that build off of what we already know.
Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.
I hope that you have learned not only how to complete the square, but also how to appreciate the idea behind the concept. Hopefully this method can save you lots of time on the next math problem you encounter!