The document discusses factoring quadratic trinomials and polynomials that are not easily factored. It provides examples of factoring quadratic expressions using the factors of ac and the quadratic formula. In one example, it factors the expression m^2 + 2m - 6399 by first finding the roots with the quadratic formula, then using the factor theorem to write the factored form as (m - 79)(m + 81). In another example, it factors the expression 20x^2 - 53x + 12 by finding two terms whose sum is the middle term, then grouping the expression accordingly.

1. Section 11-6
Factoring Quadratic Trinomials and Related
Polynomials
Sunday, March 15, 2009

2. What do you do for
quadratics that are
not that easy to
factor?
Sunday, March 15, 2009

3. What do you do for
quadratics that are
not that easy to
factor?
Sunday, March 15, 2009

4. What do you do for
quadratics that are
not that easy to
factor?
2
−b ± b − 4ac
x=
2a
Sunday, March 15, 2009

5. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Sunday, March 15, 2009

6. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
Sunday, March 15, 2009

7. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399
Sunday, March 15, 2009

8. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399 -1 + 6399 ≠ 2
Sunday, March 15, 2009

9. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399 -1 + 6399 ≠ 2
(-3)(2133) = -6399
Sunday, March 15, 2009

10. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399 -1 + 6399 ≠ 2
(-3)(2133) = -6399 -3 + 2133 ≠ 2
Sunday, March 15, 2009

11. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399 -1 + 6399 ≠ 2
(-3)(2133) = -6399 -3 + 2133 ≠ 2
...this could take a while. What can we do?
Sunday, March 15, 2009

12. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399 -1 + 6399 ≠ 2
(-3)(2133) = -6399 -3 + 2133 ≠ 2
...this could take a while. What can we do?
Sunday, March 15, 2009

13. Example 1: Solve by
factoring.
m2 + 2m - 6399 = 0
Okay, so ac is -6399. What are the factors of this?
(-1)(6399) = -6399 -1 + 6399 ≠ 2
(-3)(2133) = -6399 -3 + 2133 ≠ 2
...this could take a while. What can we do?
2
−b ± b − 4ac
x=
2a
Sunday, March 15, 2009

14. 2
−2 ± 2 − 4(1)(−6399)
m=
2
€
Sunday, March 15, 2009

15. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€
€
Sunday, March 15, 2009

16. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
€
€
Sunday, March 15, 2009

17. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
€
Sunday, March 15, 2009

18. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
€
Sunday, March 15, 2009

19. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
€
Sunday, March 15, 2009

20. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
−2 + 160
€= 2
Sunday, March 15, 2009

21. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
−2 + 160 −2 −160
€= =
2 2
Sunday, March 15, 2009

22. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
−2 + 160 −2 −160
€= =
2 2
Sunday, March 15, 2009

23. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
−2 + 160 −2 −160
€= =
79 2 2
Sunday, March 15, 2009

24. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
−2 + 160 −2 −160
€= =
79 2 2
Sunday, March 15, 2009

25. 2
−2 ± 2 − 4(1)(−6399)
m=
2
−2 ± 4 + 25596
=
2
€ −2 ± 25600
=
2
−2 ± 160
€
=
2
€
−2 + 160 −2 −160
€= =
79 −81
2 2
Sunday, March 15, 2009

26. So, we know the roots are 79 and -81. We still need
to factor this!
Sunday, March 15, 2009

27. So, we know the roots are 79 and -81. We still need
to factor this!
The Factor Theorem tells us that if we know a root r,
then the factor will be x - r.
Sunday, March 15, 2009

28. So, we know the roots are 79 and -81. We still need
to factor this!
The Factor Theorem tells us that if we know a root r,
then the factor will be x - r.
m2 + 2m - 6399 = 0
Sunday, March 15, 2009

29. So, we know the roots are 79 and -81. We still need
to factor this!
The Factor Theorem tells us that if we know a root r,
then the factor will be x - r.
m2 + 2m - 6399 = 0
(m - 79)
Sunday, March 15, 2009

30. So, we know the roots are 79 and -81. We still need
to factor this!
The Factor Theorem tells us that if we know a root r,
then the factor will be x - r.
m2 + 2m - 6399 = 0
(m - 79)(m + 81)
Sunday, March 15, 2009

31. So, we know the roots are 79 and -81. We still need
to factor this!
The Factor Theorem tells us that if we know a root r,
then the factor will be x - r.
m2 + 2m - 6399 = 0
(m - 79)(m + 81) = 0
Sunday, March 15, 2009

32. Example 2: Factor.
Sunday, March 15, 2009

33. Example 2: Factor.
20x2 - 53x + 12
Sunday, March 15, 2009

34. Example 2: Factor.
20x2 - 53x + 12 (20)(12) =
Sunday, March 15, 2009

35. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
Sunday, March 15, 2009

36. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
(-40)(-6) = 240
Sunday, March 15, 2009

37. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
(-40)(-6) = 240
-40 - 6 ≠ -53
Sunday, March 15, 2009

38. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
(-40)(-6) = 240
-40 - 6 ≠ -53
(-48)(-5) = 240
Sunday, March 15, 2009

39. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
(-40)(-6) = 240
-40 - 6 ≠ -53
(-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

40. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
-40 - 6 ≠ -53
(-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

41. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
(-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

42. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x (-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

43. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x (-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

44. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) (-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

45. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12 (-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

46. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x (-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

47. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
-48 - 5 = -53
Sunday, March 15, 2009

48. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) -48 - 5 = -53
Sunday, March 15, 2009

49. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
Sunday, March 15, 2009

50. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
Sunday, March 15, 2009

51. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula:
Sunday, March 15, 2009

52. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
€
Sunday, March 15, 2009

53. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960
=
40 €
Sunday, March 15, 2009

54. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960 53 ± 1849
= =
40 40
€
Sunday, March 15, 2009

55. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960 53 ± 1849 53 ± 43
=
= =
40
40 40
€
Sunday, March 15, 2009

56. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960 53 ± 1849 53 ± 43
=
= =
40
40 40
€
96
=
40
Sunday, March 15, 2009

57. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960 53 ± 1849 53 ± 43
=
= =
40
40 40
€
96 10
= =
40 40
Sunday, March 15, 2009

58. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960 53 ± 1849 53 ± 43
=
= =
40
40 40
€
96 12 10
= = =
40 5 40
Sunday, March 15, 2009

59. Example 2: Factor.
20x2 - 53x + 12 (20)(12) = 240
20x2 - 5x - 48x + 12 (-40)(-6) = 240
(20x2 - 5x) + (- 48x + 12) -40 - 6 ≠ -53
5x(4x - 1) - 12(4x - 1) (-48)(-5) = 240
(4x - 1) (5x - 12) -48 - 5 = -53
By Quadratic Formula: 2
53 ± (−53) − 4(20)(12)
x=
2(20)
53 ± 2809 − 960 53 ± 1849 53 ± 43
=
= =
40
40 40
€
96 12 10 1
= = = =
40 5 40 4
Sunday, March 15, 2009

60. 12 1
= =
5 4
€ €
Sunday, March 15, 2009

61. 12 1
= =
5 4
(x − 5 )
12
€ €
€
Sunday, March 15, 2009

62. 12 1
= =
5 4
(x − 5 ) (x − 4 )
12 1
€ €
€ €
Sunday, March 15, 2009

63. 12 1
= =
5 4
(x − 5 ) (x − 4 )
12 1
5( x − )
12
5
€ €
€ €
€
Sunday, March 15, 2009

64. 12 1
= =
5 4
(x − 5 ) (x − 4 )
12 1
5( x − ) 4( x − )
12 1
5 4
€ €
€ €
€ €
Sunday, March 15, 2009

65. 12 1
= =
5 4
(x − 5 ) (x − 4 )
12 1
5( x − ) 4( x − )
12 1
5 4
€ €
€ €
(5x −12)
€ €
€
Sunday, March 15, 2009

66. 12 1
= =
5 4
(x − 5 ) (x − 4 )
12 1
5( x − ) 4( x − )
12 1
5 4
€ €
€ €
(5x −12) (4 x −1)
€ €
€ €
Sunday, March 15, 2009

67. Example 3: Factor
over irrational roots.
4y2 - 6
Sunday, March 15, 2009

68. Example 3: Factor
over irrational roots.
4y2 - 6
This looks like a difference of squares, only 6 isn’t a
perfect square.
Sunday, March 15, 2009

69. Example 3: Factor
over irrational roots.
4y2 - 6
This looks like a difference of squares, only 6 isn’t a
perfect square...or is it?
Sunday, March 15, 2009

70. Example 3: Factor
over irrational roots.
4y2 - 6
This looks like a difference of squares, only 6 isn’t a
perfect square...or is it?
(2y −
€
Sunday, March 15, 2009

71. Example 3: Factor
over irrational roots.
4y2 - 6
This looks like a difference of squares, only 6 isn’t a
perfect square...or is it?
(2y − (2y +
€ €
Sunday, March 15, 2009

72. Example 3: Factor
over irrational roots.
4y2 - 6
This looks like a difference of squares, only 6 isn’t a
perfect square...or is it?
(2y − 6) (2y +
€€
€
Sunday, March 15, 2009

73. Example 3: Factor
over irrational roots.
4y2 - 6
This looks like a difference of squares, only 6 isn’t a
perfect square...or is it?
(2y − 6) (2y + 6)
€€ €
€
Sunday, March 15, 2009

74. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
Sunday, March 15, 2009

75. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D=
Sunday, March 15, 2009

76. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5)
Sunday, March 15, 2009

77. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20
Sunday, March 15, 2009

78. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
Sunday, March 15, 2009

79. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
Sunday, March 15, 2009

80. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124
x=
2
Sunday, March 15, 2009

81. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31
x= =
2 2
€
Sunday, March 15, 2009

82. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31 12 ± 2 31
x= = =
2 2 2
€ €
Sunday, March 15, 2009

83. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31 12 ± 2 31
x= = = = 6 ± 31
2 2 2
€
€ €
Sunday, March 15, 2009

84. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31 12 ± 2 31
x= = = = 6 ± 31
2 2 2
(x −
€
€ €
Sunday, March 15, 2009

85. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31 12 ± 2 31
x= = = = 6 ± 31
2 2 2
(x − (x €
−
€ €
Sunday, March 15, 2009

86. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31 12 ± 2 31
x= = = = 6 ± 31
2 2 2
(x − 6 − 31) (x −
€
€ €
Sunday, March 15, 2009

87. Example 4: Factor
over irrational zeros.
x2 - 12x + 5
D = (-12)2 - 4(1)(5) = 144 - 20 = 124
This time, we’ll need the quadratic formula.
12 ± 124 12 ± 4 • 31 12 ± 2 31
x= = = = 6 ± 31
2 2 2
(x − 6 − 31) (x − 6 + 31)
€
€ €
Sunday, March 15, 2009

88. Example 5: Solve.
12x3 + 18x2 + 12x = 0
Sunday, March 15, 2009

89. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x
Sunday, March 15, 2009

90. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2
Sunday, March 15, 2009

91. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x
Sunday, March 15, 2009

92. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
Sunday, March 15, 2009

93. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D=
Sunday, March 15, 2009

94. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2)
Sunday, March 15, 2009

95. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16
Sunday, March 15, 2009

96. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16 = -7
Sunday, March 15, 2009

97. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16 = -7
−3 ± −7
x=
4
€
Sunday, March 15, 2009

98. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16 = -7
−3 ± −7 −3 ± i 7
x= =
4 4
€ €
Sunday, March 15, 2009

99. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16 = -7
−3 ± −7 −3 ± i 7 7
3
i
=− ±
x= = 4 4
4 4
€ €
€
Sunday, March 15, 2009

100. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16 = -7
−3 ± −7 −3 ± i 7 7
3
i
=− ±
x= = 4 4
4 4
7
3
x =− ± i
4 4
€ €
€
€
Sunday, March 15, 2009

101. Example 5: Solve.
12x3 + 18x2 + 12x = 0
6x(2x2 + 3x + 2) = 0
D = 32 - 4(2)(2) = 9 - 16 = -7
−3 ± −7 −3 ± i 7 7
3
i
=− ±
x= = 4 4
4 4
7
3
x =− ± i
4 4
AND x = 0
€ €
€
€
Sunday, March 15, 2009

102. Homework:
Sunday, March 15, 2009

103. Homework:
p. 709 #1-24, skip #21
“Learn from yesterday, live for today, hope for tomorrow.
The important thing is not to stop questioning.” – Albert
Einstein
Sunday, March 15, 2009