Perfect Squares and Difference of Squares

1. SECTION 9-8
Perfect Squares and
Differences of Squares

2. Essential Questions
How do you factor perfect square trinomials?
How do you factor the difference of perfect squares?
Where you’ll see this:
Travel, number sense, modeling, geography

3. Vocabulary
1. Perfect Square Trinomial:
2. Difference of Two Squares:

4. Vocabulary
1. Perfect Square Trinomial: A trinomial that can be factored
into a binomial squared
2. Difference of Two Squares:

5. Vocabulary
1. Perfect Square Trinomial: A trinomial that can be factored
into a binomial squared
2. Difference of Two Squares: A polynomial that can be factored
into two binomials with the same terms but different signs
in between

6. Perfect Square Trinomial
2
( a − b)

7. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)

8. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)
2 2
a − ab − ab + b

9. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)
2 2
a − ab − ab + b
2 2
a − 2ab + b

10. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)
2 2
a − ab − ab + b
2 2
a − 2ab + b

11. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)
2 2
a − ab − ab + b
2 2
a − 2ab + b

12. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)
2 2
a − ab − ab + b
2 2
a − 2ab + b

13. Perfect Square Trinomial
2
( a − b)
( a − b)( a − b)
2 2
a − ab − ab + b
2 2
a − 2ab + b
First term squared, last term squared, middle term is
2 times first term times last term

14. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
2 2
c. x − 14x + 49 d. x + 7x + 14

15. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
(3x
2 2
c. x − 14x + 49 d. x + 7x + 14

16. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
2
(3x +2)
2 2
c. x − 14x + 49 d. x + 7x + 14

17. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
2
(3x +2) (y
2 2
c. x − 14x + 49 d. x + 7x + 14

18. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
2 2
(3x +2) ( y −10)
2 2
c. x − 14x + 49 d. x + 7x + 14

19. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
2 2
(3x +2) ( y −10)
2 2
c. x − 14x + 49 d. x + 7x + 14
2
( x − 7)

20. Example 1
Factor.
2 2
a. 9x + 12x + 4 b. y − 20 y + 100
2 2
(3x +2) ( y −10)
2 2
c. x − 14x + 49 d. x + 7x + 14
2
( x − 7) Not a perfect square trinomial

21. Difference of Squares
( a − b)( a + b)

22. Difference of Squares
( a − b)( a + b)
2 2
a + ab − ab − b

23. Difference of Squares
( a − b)( a + b)
2 2
a + ab − ab − b
2 2
a −b

24. Difference of Squares
( a − b)( a + b)
2 2
a + ab − ab − b
2 2
a −b

25. Difference of Squares
( a − b)( a + b)
2 2
a + ab − ab − b
2 2
a −b

26. Difference of Squares
( a − b)( a + b)
2 2
a + ab − ab − b
2 2
a −b
First term squared, last term squared, subtraction between
the two terms

27. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
2 2
c. t − 900 d. y + 100
4 4 8 2
e. 25x y − 36z f. 16h − 144

28. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x
2 2
c. t − 900 d. y + 100
4 4 8 2
e. 25x y − 36z f. 16h − 144

29. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9)
2 2
c. t − 900 d. y + 100
4 4 8 2
e. 25x y − 36z f. 16h − 144

30. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9)
2 2
c. t − 900 d. y + 100
4 4 8 2
e. 25x y − 36z f. 16h − 144

31. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11)
2 2
c. t − 900 d. y + 100
4 4 8 2
e. 25x y − 36z f. 16h − 144

32. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
4 4 8 2
e. 25x y − 36z f. 16h − 144

33. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30)
4 4 8 2
e. 25x y − 36z f. 16h − 144

34. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30) (t − 30)
4 4 8 2
e. 25x y − 36z f. 16h − 144

35. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30) (t − 30) Not a difference of squares
4 4 8 2
e. 25x y − 36z f. 16h − 144

36. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30) (t − 30) Not a difference of squares
4 4 8 2
e. 25x y − 36z f. 16h − 144
2 2 4
(5x y + 6z )

37. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30) (t − 30) Not a difference of squares
4 4 8 2
e. 25x y − 36z f. 16h − 144
2 2 4 2 2 4
(5x y + 6z ) (5x y − 6z )

38. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30) (t − 30) Not a difference of squares
4 4 8 2
e. 25x y − 36z f. 16h − 144
2 2 4 2 2 4
(5x y + 6z ) (5x y − 6z ) (4h + 12)

39. Example 2
Factor.
2 2
a. 64x − 81 b. r − 121
(8x −9) (8x + 9) (r + 11) (r − 11)
2 2
c. t − 900 d. y + 100
(t + 30) (t − 30) Not a difference of squares
4 4 8 2
e. 25x y − 36z f. 16h − 144
2 2 4 2 2 4
(5x y + 6z ) (5x y − 6z ) (4h + 12) (4h − 12)

40. Problem Set

41. Problem Set
p. 410 #1-42 multiples of 3
“There are two kinds of men who never amount to much:
those who cannot do what they are told and those who
can do nothing else.” - Cyrus H. Curtis