Determine the number that will complete the square to solve each equation, after the constant
term has been written on the right side and the coefficient of the second-degree term is 1. Do
not actually solve. See Examples 6 – 8.
Solve each equation by completing the square. Use the results of Exercises 49 –54to solve
Exercises 57– 62.See Examples 6 – 8.
- 72.p^2 -
8
3
z^2 - p=- 1
4
3
z=-
1
9
25 n^2 - 20 n= 1
5 x^2 - 10 x+ 2 = 0 2 x^2 - 16 x+ 25 = 0 9 x^2 - 24 x=- 13
x^2 + 13 x- 3 = 0 2 k^2 + 5 k- 2 = 0 3 r^2 + 2 r- 2 = 0
3 w^2 - w= 24 4 z^2 - z= 39 x^2 + 7 x- 1 = 0
t^2 + 2 t- 1 = 0 x^2 + 10 x+ 18 = 0 x^2 + 8 x+ 11 = 0
x^2 - 2 x- 24 = 0 m^2 - 4 m- 32 = 0 x^2 + 4 x- 2 = 0
x^2 + 8 x+ 11 = 0 3 w^2 - w- 24 = 0 4 z^2 - z- 39 = 0
x^2 + 4 x- 2 = 0 t^2 + 2 t- 1 = 0 x^2 + 10 x+ 18 = 0
504 CHAPTER 9 Quadratic Equations, Inequalities, and Functions
73.
(Hint: First clear the decimals.)
74.
(Hint: First clear the decimals.)
0.1x^2 - 0.2x-0.1= 0 0.1p^2 - 0.4p+0.1= 0
Find the nonreal complex solutions of each equation. See Example 9.
- 4 x^2 + 5 x+ 5 = 0 85.-m^2 - 6 m- 12 = 0 86.-x^2 - 5 x- 10 = 0
m^2 + 4 m+ 13 = 0 t^2 + 6 t+ 10 = 0 3 r^2 + 4 r+ 4 = 0
1 t+ 622 =- 9 16 x- 122 =- 8 14 m- 722 =- 27
x^2 =- 12 x^2 =- 18 1 r- 522 =- 4
EXERCISES 87– 92
FOR INDIVIDUAL OR GROUP WORK
The Greeks had a method of completing the square geometrically in which they literally
changed a figure into a square. For example, to complete the square for , we
begin with a square of side x, as in the figure on the left. We add three rectangles of
width 1 to the right side and the bottom to get a region with area. To fill in the
corner (complete the square), we must add nine 1 -by- 1 squares as shown.
Work Exercises 87–92 in order.
87.What is the area of the original square?
88.What is the area of each strip?
89.What is the total area of the six strips?
90.What is the area of each small square in the corner of the second figure?
91.What is the total area of the small squares?
92.What is the area of the new “complete” square?
x^2 + 6 x
x^2 + 6 x
RELATING CONCEPTS
x x + 3
x + 3
x
