4-6 Completing the Square

Quick Review

If you cannot solve a quadratic equation by factoring, you can use completing the square. You write one side as a perfect square trinomial and then take square roots. You can also convert a quadratic function from standard form to vertex form by completing the square.

Example

Solve x 2 + 6 x − 7 = 0 x squared , plus 6 x minus 7 equals 0  by completing the square.

Exercises

Solve each equation by finding square roots.

49. 4 x 2 = 16 4 x squared , equals 16
50. 4 x 2 − 20 = 0 4 x squared , minus 20 equals 0
51. 5 x 2 − 45 = 0 5 x squared , minus 45 equals 0
52. 3 x 2 = 36 3 x squared , equals 36

What values complete each square?

53. x 2 − 6 x x squared , minus 6 x
54. x 2 + 3 x x squared , plus 3 x

Solve each equation by completing the square.

55. x 2 + 8 x + 6 = 0 x squared , plus 8 x plus 6 equals 0
56. x 2 − 10 x = 13 x squared , minus 10 x equals 13
57. 9 x 2 + 6 x + 1 = 4 9 x squared , plus 6 x plus 1 equals 4
58. x 2 − 2 x + 4 = 0 x squared , minus 2 x plus 4 equals 0
59. x 2 + 3 x = − 25 x squared , plus 3 x equals negative 25
60. 4 x 2 − x − 3 = 0 4 x squared , minus x minus 3 equals 0

4-7 The Quadratic Formula

Quick Review

You can solve a quadratic equation in the form a x 2 + b x + c = 0 eh , x squared , plus b x plus c equals 0  by using the Quadratic Formula, x = − b ± b 2 − 4 a c 2 a . x equals . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction . .  The discriminant of a quadratic equation in standard form is the value of the expression b 2 − 4 a c . b squared , minus 4 eh c .  You can use it to find the quantity and type of solutions of a quadratic equation.

Example

Use the Quadratic Formula to solve 2 x 2 − 6 x = − 3 . 2 x squared , minus 6 x equals negative 3 .

Exercises

Solve each equation using the quadratic formula.

61. 3 x 2 + 5 x = 8 3 x squared , plus 5 x equals 8
62. x 2 = 6 x − 9 x squared , equals 6 x minus 9
63. x ( x − 3 ) = 4 x open x minus 3 close equals 4
64. 5 x 2 − 7 x − 3 = 0 5 x squared , minus 7 x minus 3 equals 0

Determine the discriminant of each equation. How many real solutions does each equation have?

65. 4 x 2 − 2 x = 10 4 x squared , minus 2 x equals 10
66. x 2 − 5 x + 7 = 0 x squared , minus 5 x plus 7 equals 0
67. 3 x 2 + 3 = 6 x 3 x squared , plus 3 equals 6 x
68. 7 − 3 x = 8 x 2 7 minus 3 x equals , 8 x squared
69. Gardening Margaret is planning a rectangular garden. Its length is 4 ft less than twice its width. Its area is 170 ft 2 . 170 , ft squared , .  What are the dimensions of the garden?

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