© D. T. Simmons, 2009 1

Completing the Square

Factoring “unfactorable” 2nd degree trinomials

Don Simmons

© D. T. Simmons, 2009 2

Completing the Square

• We have learned earlier that a perfect square trinomial can always be factored.

• Therefore, if we have a trinomial we cannot factor using integers, we can change it in such a way that we are dealing with a perfect square trinomial.

© D. T. Simmons, 2009 3

Completing the Square

• Recall that a perfect square trinomial is always in the form:

• Therefore, we have to change the polynomial so that it fits the form.

• To get the most out of this presentation, use pencil and paper and work through the instructions slowly and carefully.

22 2 baba

© D. T. Simmons, 2009 4

Completing the Square

The equation we are going to solve is the following…

By testing whether or not the factors of c can sum to equal b, we can determine if the trinomial is factorable. This trinomial is not factorable in its present form.

2

2 16 20 0x x

Step 1

Divide by the leading coefficient to set the a-value to 1.

© D. T. Simmons, 2009 5

2

2

2 16 20 02

8 10 0

x x

x x

Step 2

Re-write the equation in the form ax + by = c

© D. T. Simmons, 2009 6

2

2

2

8 10 0

8 10 10 10

18 0

0

x x

x x

x x

Step 3

Find one-half of the b value.

Add the square of that number to both sides.

© D. T. Simmons, 2009 7

2

2

2

2 2

8 10

8 104 4

16 28 6

x x

x x

x x

Step 4

Re-write the perfect square trinomial as a binomial squared.

Find the square root of each side of the equation.

© D. T. Simmons, 2009 8

2

24

8 16 26

26

264

x

x

x x

Step 5

Solve for x.

© D. T. Simmons, 2009 9

4 2

4

2

6

264

6

4

4

x

x

x

© D. T. Simmons, 2009 10

Try it. You’ll like it!

That’s all folks!