Completing the Square

Today I would like to discuss the method of completing the square. This is a technique with many useful applications. In particular, there are several types of problems on the revised SAT where this technique can be used.

We complete the square on an expression of the form

x2bx

To complete the square we simply take half of b, and then square the result. In other words we get (b/2)2.

The expression x2bx + (b/2)2 is always a perfect square. In fact,

x2bx + (b/2)= (xb/2)2

Example 1: Let’s complete the square in the expression x2 + 6x.

Well half of 6 is 3, and when we square 3 we get 9. So the new expression is x2 + 6+ 9 which factors as (x + 3)2.

Important notes: (1) When we complete the square we usually get an expression that is not equal to the original expression. For example,

x2 + 6x ≠ x26+ 9.

(2) The coefficient of x2 must be 1 before we complete the square. So, for example, we cannot complete the square on the expression

2x2 + 32x.

But we can first factor out the 2 to get 2(x2 + 16x), and then complete the square on the expression x2 + 16x to get

2(x2 + 16x + 64).

Note that we increased the expression by 2 ⋅ 64 = 128 .

Example 2:

x2 – 8x

If the method of completing the square is used to rewrite the expression above in the form (x – h)2 + k, then what is the value of h – k ?

I will provide you with the solution to this problem next week. In the meantime feel free to try this problem yourself and post your own solution in the comments.