Completing The Square

Completing The Square is a method for factoring and solving quadratic equations. In fact, by using the Completing The Square method, we can derive a general equation for solving quadratics, the familiar “Quadratic Formula”.

Let’s look at some examples first. We need to be able to recognize a perfect square when we see it. Let’s try (x+1)^{2}. (x+1)^{2} = x^{2} + 2x + 1. If you are unfamiliar with the “FOIL” method, or how to multiply polynomials, make sure to visit the lesson on factoring before you continue with this section.

(x + 2)^{2} = x^{2} + 4x + 4, (x + 3)^{2} = x^{2} + 6x + 9, (x + 4)^{2} = x^{2} + 8x + 16, (x + 5)^{2} = x^{2} + 10x + 25

We notice a pattern here. The constant term is always one-half the middle term squared. Take a look at these examples until you see the pattern clearly. This gives us the basis for our completing the square method.

So, if we want to factor x^{2} + 6x + 9, we recognize the pattern and see that it is (x + 3)^{2}. Polynomials that follow this pattern are known as “Perfect Squares”. But what if we want to factor a polynomial like x^{2} + 8x + 12? It is not a perfect square, so we need another way to solve this.

We use our pattern to create a perfect square: x^{2} + 8x + 12. Well, we know that in a perfect square, the constant term is one-half the middle term squared. In other words, ½ of 8 is 4, and 4 squared is 16. Thus, x^{2} + 8x + 16 is a perfect square.

Here is the trick: We want a 16 in our expression, but we don’t want to change the value of it. So we will *add 16 and subtract it*.

In other words: x^{2} + 8x + 12 = x^{2} + 8x + 12 ( + 16 - 16), and re-arrange it: x^{2} + 8x + 16 ( + 12 - 16). We re-group the expression, moving the 16 with the rest of the polynomial and moving the 12 to the end. So we now have: x^{2} + 8x +16 (- 4), and x^{2} + 8x + 16 is a perfect square which we can factor, so we now have:

x^{2} + 8x + 12 = (x + 4)^{2} – 4. Incidentally, this form of polynomial is called the “Vertex Form.”

The equation for the general quadratic is: ax^{2} + bx + c = f(x)

If we complete the square for this, we add/subtract _{} to the expression. When we use some algebra to re-arrange this new expression, we get the familiar equation for solving quadratics:

_{}