1. 9.6 Perfect Squares & Factoring

2. Perfect Square Trinomials
First term must be a perfect square
Last term must be a perfect square
Middle term must equal 2(first term)(last term)
Then, a2+2ab+b2=(a+b)2
and a2-2ab+b2=(a+b)2

3. Determine whether is a perfect square trinomial. If so, factor it.
1. Is the first term a perfect square?
Yes,
2. Is the last term a perfect square?
Yes,
3. Is the middle term equal to ?
Yes,
Answer: is a perfect square trinomial.
Write as
Factor using the pattern.
Example 6-1a

4. Determine whether is a perfect square trinomial. If so, factor it.
1. Is the first term a perfect square?
Yes,
2. Is the last term a perfect square?
Yes,
3. Is the middle term equal to ?
No,
Answer: is not a perfect square trinomial.
Example 6-1a

5. Answer: not a perfect square trinomial
Determine whether each trinomial is a perfect square trinomial. If so, factor it. a. b.
Answer: not a perfect square trinomial
Answer: yes;
Example 6-1b

6. Factor .
First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares.
6 is the GCF.
and
Factor the difference of squares.
Answer:
Example 6-2a

7. Factor .
This polynomial has three terms that have a GCF of 1. While the first term is a perfect square, the last term is not. Therefore, this is not a perfect square trinomial.
This trinomial is in the form Are there two numbers m and n whose product is and whose sum is 8? Yes, the product of 20 and –12 is –240 and their sum is 8.
Example 6-2a

8. Answer: Write the pattern. and Group terms with common factors.
Factor out the GCF from each grouping.
is the common factor.
Answer:
Example 6-2a

9. Factor each polynomial. a.b.
Answer:
Answer:
Example 6-2b

10. Solving Perfect Square Trinomials

11. Recognize as a perfect square trinomial.
Solve
Original equation
Recognize as a perfect square trinomial.
Factor the perfect square trinomial.
Set the repeated factor equal to zero.
Solve for x.
Answer: Thus, the solution set is Check this
solution in the original equation.
Example 6-3a

12. Solve
Answer:
Example 6-3b

13. Separate into two equations. or
Solve .
Original equation
Square Root Property
Add 7 to each side.
Separate into two equations. or
Simplify.
Answer: The solution set is Check each solution in the original equation.
Example 6-4a

14. Recognize perfect square trinomial.
Solve .
Original equation
Recognize perfect square trinomial.
Factor perfect square trinomial.
Square Root Property
Subtract 6 from each side.
Example 6-4a

15. Separate into two equations.or
Separate into two equations.
Simplify.
Answer: The solution set is Check this solution in the original equation.
Example 6-4a

16. Subtract 9 from each side.
Solve .
Original equation
Square Root Property
Subtract 9 from each side.
Answer: Since 8 is not a perfect square, the solution set is
Using a calculator, the approximate
solutions are or about –6.17 and
or about –11.83.
Example 6-4a

17. Check You can check your answer using a graphing calculator. Graph and Using the INTERSECT feature of your graphing calculator, find where The check of –6.17 as one of the approximate solutions is shown.
Example 6-4a

18. Solve each equation. Check your solutions. a.b c.
Answer:
Answer:
Answer:
Example 6-4b