Solving Quadratics by Completing the Square

Some quadratics are easy to factor, such as x^2 + 4x + 4, which turns out to be (x+2)^2. But say you have the equation x^2 + 4x + 2 = 0. Obviously, you can not factor that. However, you can factor x^2 + 4x + 4, as shown above. So basically, we need to add 2 to both sides of the equation in order to make it factorable. This is what completing the square is. You take a non-factorable quadratic and try to add a value that will make it a perfect square trinomial (one that can be factored into a value squared, like the one above)

Why a perfect square trinomial, you ask? Because the zero-product property does not work in this case. The factors could be anything. If the trinomial is a perfect square, then there are only two options for the number: it and its opposite. Therefore, the expression on the right side has a +- (plus or minus) after the square root is taken.

Let's work on that original problem. We have x^2 + 4x + 2 = 0. Now, we need to make the left side a perfect square. To do this, we add 2 to both sides, as mentioned. This results in x^2 + 4x + 4 = 2. Now, we factor and simplify.

(x+2)^2 = 2
 x+2    = +-sqrt(2)         
 x      = -2 +- sqrt(2)     

However, this will not work for a-coefficients greater than 1 (except for the perfect squares). We will have to divide the trinomial by the a-coefficient to set it up.

For an example, take 2x^2 + 12x + 7 = 0. The completing the square method is then as follows.

2x^2 + 12x + 7  = 0
x^2  + 6x + 3.5 = 0     Divide by 2
x^2  + 6x + 9   = 5.5   Perfect square is x^2 + 6x + 9
(x + 3)^2       = 5.5
 x + 3          = +-sqrt(5.5)
 x              = -3 +- sqrt(5.5)

However, if we were given 4x^2 + 8x + 1 = 0, we could just add 1 and simplify to get (2x + 2)^2 = 1. (Note: Since the answer here is a whole number, the trinomial is, indeed, factorable)


Some extra points:

• The completing the square method will always work. However, if it gets really messy, just use the quadratic formula.


• You do not need to guess on the number to add. Once you set up the equation in the form x^2 + bx = c, the number to add is (b/2)^2. This is based off the middle term and the last term of the (a+b)^2 expansion.