9-5 Completing the Square

Objective

To solve quadratic equations by completing the square

Image Long Description

In previous lessons, you solved quadratic equations by finding square roots and by factoring. These methods work in some cases, but not all.

Essential Understanding You can solve any quadratic equation by first writing it in the form m 2 = n . m squared , equals n .

You can model this process using algebra tiles. The algebra tiles below represent the expression x 2 + 8 x . x squared , plus 8 x .

Here is the same expression rearranged to form part of a square. Notice that the x-tiles have been split evenly into two groups of four.

You can complete the square by adding 4 2 , 4 squared , comma  or 16, 1-tiles. The completed square is x 2 + 8 x + 16 , x squared , plus 8 . x plus 16 comma  or ( x + 4 ) 2 . open x plus 4 close squared . .

In general, you can change the expression x 2 + b x x squared , plus , b x  into a perfect-square trinomial by adding ( b 2 ) 2 open , b over 2 , close squared  to x 2 + b x . x squared , plus , b x .  This process is called completing the square. The process is the same whether b is positive or negative.

Table of Contents