Solving Quadratic Equations by Completing the Square

1. Section 3-5
Solving Quadratic Equations by Completing the
Square

2. Essential Questions
• How do you solve quadratic functions by
using the Square Root Property?
• How do you solve quadratic functions by
completing the square?

3. Vocabulary
1. Completing the Square:

4. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared

5. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared
Step 1:

6. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared
Step 1: Take half of b in the quadratic

7. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared
Step 1: Take half of b in the quadratic
Step 2:

8. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared
Step 1: Take half of b in the quadratic
Step 2: Square the result of Step 1

9. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared
Step 1: Take half of b in the quadratic
Step 2: Square the result of Step 1
Step 3:

10. Vocabulary
1. Completing the Square: The method used to
turn any quadratic into a perfect square
trinomial for the purpose of factoring into a
binomial squared
Step 1: Take half of b in the quadratic
Step 2: Square the result of Step 1
Step 3: Add the result of Step 2 to
complete a perfect square trinomial
(add to both sides if in an equation)

11. Vocabulary
2. Vertex Form of a Quadratic:

12. Vocabulary
2. Vertex Form of a Quadratic: When completing
the square, the resulting form will provide
the vertex of the quadratic

13. Vocabulary
2. Vertex Form of a Quadratic: When completing
the square, the resulting form will provide
the vertex of the quadratic
y = a(x − h)2
+ k

14. Vocabulary
2. Vertex Form of a Quadratic: When completing
the square, the resulting form will provide
the vertex of the quadratic
y = a(x − h)2
+ k
The vertex is given by (h, k)

15. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64

16. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64

17. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64
(x + 7)2
= ± 64

18. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64
(x + 7)2
= ± 64
x + 7 = ±8

19. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64
(x + 7)2
= ± 64
x + 7 = ±8
x + 7 = 8

20. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64
(x + 7)2
= ± 64
x + 7 = ±8
x + 7 = 8 x + 7 = −8

21. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64
(x + 7)2
= ± 64
x + 7 = ±8
x + 7 = 8 x + 7 = −8
x = 1

22. Example 1
Solve by using the Square Root Property.
x 2
+14x + 49 = 64
(x + 7)2
= 64
(x + 7)2
= ± 64
x + 7 = ±8
x + 7 = 8 x + 7 = −8
x = 1 x = −15

23. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0

24. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0
+12+12

25. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0
+12+12
x 2
+ 4x + ____ = 12

26. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
+12+12
x 2
+ 4x + ____ = 12

27. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
+12+12
x 2
+ 4x + ____ = 12

28. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
+12+12
x 2
+ 4x + ____ = 12

29. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 12

30. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124

31. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4

32. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16

33. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16

34. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16
(x + 2)2
= ± 16

35. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16
(x + 2)2
= ± 16
x + 2 = ±4

36. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16
(x + 2)2
= ± 16
x + 2 = ±4
x + 2 = 4

37. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16
(x + 2)2
= ± 16
x + 2 = ±4
x + 2 = 4 x + 2 = −4

38. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16
(x + 2)2
= ± 16
x + 2 = ±4
x + 2 = 4 x + 2 = −4
x = 2

39. Example 2
Solve by completing the square.
x 2
+ 4x −12 = 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
+12+12
x 2
+ 4x + ____ = 124 +4
x 2
+ 4x + 4 = 16
(x + 2)2
= 16
(x + 2)2
= ± 16
x + 2 = ±4
x + 2 = 4 x + 2 = −4
x = 2 x = −6

40. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0

41. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0
+1 +1

42. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0
+1 +1
3x 2
− 2x + ____ = 1

43. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0
+1 +1
3x 2
− 2x + ____ = 1
x 2
− 2
3 x + ____ = 1
3

44. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
+1 +1
3x 2
− 2x + ____ = 1
x 2
− 2
3 x + ____ = 1
3

45. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2+1 +1
3x 2
− 2x + ____ = 1
x 2
− 2
3 x + ____ = 1
3

46. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
+1 +1
3x 2
− 2x + ____ = 1
x 2
− 2
3 x + ____ = 1
3

47. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
x 2
− 2
3 x + ____ = 1
3

48. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
x 2
− 2
3 x + ____ = 1
3
1
9

49. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9x 2
− 2
3 x + ____ = 1
3
1
9

50. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
x 2
− 2
3 x + ____ = 1
3
1
9

51. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
x 2
− 2
3 x + ____ = 1
3
1
9

52. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
(x − 1
3 )2
= ± 4
9
x 2
− 2
3 x + ____ = 1
3
1
9

53. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
(x − 1
3 )2
= ± 4
9
x − 1
3 = ± 2
3
x 2
− 2
3 x + ____ = 1
3
1
9

54. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
(x − 1
3 )2
= ± 4
9
x − 1
3 = ± 2
3
x − 1
3 = 2
3
x 2
− 2
3 x + ____ = 1
3
1
9

55. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
(x − 1
3 )2
= ± 4
9
x − 1
3 = ± 2
3
x − 1
3 = 2
3 x − 1
3 = − 2
3
x 2
− 2
3 x + ____ = 1
3
1
9

56. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
(x − 1
3 )2
= ± 4
9
x − 1
3 = ± 2
3
x − 1
3 = 2
3 x − 1
3 = − 2
3
x = 1
x 2
− 2
3 x + ____ = 1
3
1
9

57. Example 3
Solve by completing the square.
3x 2
− 2x −1= 0 (1
2 b)2
(1
2 i 2
3 )2
(1
3 )2
1
9
+1 +1
3x 2
− 2x + ____ = 1
+ 1
9
x 2
− 2
3 x + 1
9 = 4
9
(x − 1
3 )2
= 4
9
(x − 1
3 )2
= ± 4
9
x − 1
3 = ± 2
3
x − 1
3 = 2
3 x − 1
3 = − 2
3
x = 1 x = − 1
3
x 2
− 2
3 x + ____ = 1
3
1
9

58. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0

59. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0
−11−11

60. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0
−11−11
x 2
+ 4x + ____ = −11

61. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
−11−11
x 2
+ 4x + ____ = −11

62. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2−11−11
x 2
+ 4x + ____ = −11

63. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
−11−11
x 2
+ 4x + ____ = −11

64. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
x 2
+ 4x + ____ = −11

65. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
x 2
+ 4x + ____ = −114

66. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
+4x 2
+ 4x + ____ = −114

67. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
+4
x 2
+ 4x + 4 = −7
x 2
+ 4x + ____ = −114

68. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
+4
x 2
+ 4x + 4 = −7
(x + 2)2
= −7
x 2
+ 4x + ____ = −114

69. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
+4
x 2
+ 4x + 4 = −7
(x + 2)2
= −7
(x + 2)2
= ± −7
x 2
+ 4x + ____ = −114

70. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
+4
x 2
+ 4x + 4 = −7
(x + 2)2
= −7
(x + 2)2
= ± −7
x + 2 = ±i 7
x 2
+ 4x + ____ = −114

71. Example 4
Solve by completing the square.
x 2
+ 4x +11= 0 (1
2 b)2
(1
2 i 4)2
(2)2
4
−11−11
+4
x 2
+ 4x + 4 = −7
(x + 2)2
= −7
(x + 2)2
= ± −7
x + 2 = ±i 7
x 2
+ 4x + ____ = −114
x = −2 ± i 7

72. Example 5
Write in vertex form.
−3x 2
−10x +12

73. Example 5
Write in vertex form.
−3x 2
−10x +12
−3x 2
−10x + ____+12

74. Example 5
Write in vertex form.
−3x 2
−10x +12
−3x 2
−10x + ____+12
−3(x 2
+ 10
3 x + ____)+12

75. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
−3x 2
−10x + ____+12
−3(x 2
+ 10
3 x + ____)+12

76. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2−3x 2
−10x + ____+12
−3(x 2
+ 10
3 x + ____)+12

77. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
−3x 2
−10x + ____+12
−3(x 2
+ 10
3 x + ____)+12

78. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
−3(x 2
+ 10
3 x + ____)+12

79. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
−3(x 2
+ 10
3 x + ____)+12
25
9

80. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
+ 25
3−3(x 2
+ 10
3 x + ____)+12
25
9

81. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
+ 25
3
−3(x 2
+ 10
3 x + 25
9 )+ 61
3
−3(x 2
+ 10
3 x + ____)+12
25
9

82. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
+ 25
3
−3(x 2
+ 10
3 x + 25
9 )+ 61
3
−3(x + 5
3 )2
+ 61
3
−3(x 2
+ 10
3 x + ____)+12
25
9

83. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
+ 25
3
−3(x 2
+ 10
3 x + 25
9 )+ 61
3
−3(x + 5
3 )2
+ 61
3
−3(x 2
+ 10
3 x + ____)+12
25
9
Vertex:

84. Example 5
Write in vertex form.
−3x 2
−10x +12
(1
2 b)2
(1
2 i 10
3 )2
(5
3 )2
25
9
−3x 2
−10x + ____+12
+ 25
3
−3(x 2
+ 10
3 x + 25
9 )+ 61
3
−3(x + 5
3 )2
+ 61
3
−3(x 2
+ 10
3 x + ____)+12
25
9
Vertex: (− 5
3 ,61
3 )