Intermediate Algebra (11th edition)

Use the quadratic formula to solve each equation. (All solutions for these equations are real
numbers.) See Examples 1 and 2.

23. 24. 25.

26. 27. 28.

Use the quadratic formula to solve each equation. (All solutions for these equations are non-
real complex numbers.) See Example 3.

Use the discriminant to determine whether the solutions for each equation are
A.two rational numbers B.one rational number
C.two irrational numbers D. two nonreal complex numbers.

Tell whether the equation can be solved by factoring or whether the quadratic formula should
be used. Do not actually solve. See Example 4.

Based on your answers in Exercises 39– 46,solve the equation given in each exercise.

47. Exercise 39 48. Exercise 40 49. Exercise 43 50. Exercise 44

51.Find the discriminant for each quadratic equation. Use it to tell whether the equation can
be solved by factoring or whether the quadratic formula should be used. Then solve each
equation.
(a) (b)

52.Concept Check Is it possible for the solution of a quadratic equation with integer coef-
ficients to include just one irrational number? Why or why not?

Find the value of a, b, or c so that each equation will have exactly one rational solution. See
Example 5.

59.One solution of is. Find band the other solution.

60.One solution of 3 x^2 - 7 x+c= 0 is. Find^13 cand the other solution.

4 x^2 +bx- 3 = 0 -^52

at^2 + 24 t+ 16 = 0 9 x^2 - 30 x+c= 0 4 m^2 + 12 m+c= 0

p^2 +bp+ 25 = 0 r^2 - br+ 49 = 0 am^2 + 8 m+ 1 = 0

3 x^2 + 13 x=- 12 2 x^2 + 19 = 14 x

3 m^2 - 10 m+ 15 = 0 18 x^2 + 60 x+ 82 = 0

9 x^2 - 12 x- 1 = 0 3 x^2 = 5 x+ 2 4 x^2 = 4 x+ 3

25 x^2 + 70 x+ 49 = 0 4 x^2 - 28 x+ 49 = 0 x^2 + 4 x+ 2 = 0

12 x- 1218 x- 42 =- 1 1 x- 1219 x- 32 =- 2

x 13 x+ 42 =- 2 z 12 z+ 32 =- 2

t^2 + 4 t+ 11 = 0 4 x^2 - 4 x=- 7 9 x^2 - 6 x=- 7

x^2 - 3 x+ 6 = 0 x^2 - 5 x+ 20 = 0 r^2 - 6 r+ 14 = 0

x= 12 x+ 122 =x+ 4 12 x- 122 =x+ 2

21 x+ 32

x+ 5

p=

515 - p 2

31 p+ 12

1 x+ 221 x- 32 = 1 1 x- 521 x+ 22 = 6

• 3 x 1 x+ 22 =- 4 1 r- 321 r+ 52 = 2 1 x+ 121 x- 72 = 1

p^2 + - 2 t 1 t+ 22 =- 3

p

3

=

1

6

x^2

4

-

x

2

= 1

4 r^2 - 4 r- 19 = 0 2 - 2 x= 3 x^226 r- 2 = 3 r^2

x^2 + 18 = 10 x x^2 - 4 = 2 x 4 x^2 + 4 x- 1 = 0

2 x^2 + 3 x- 1 = 0 2 x^2 - 2 x= 1 9 x^2 + 6 x= 1

x^2 - 8 x+ 15 = 0 x^2 + 3 x- 28 = 0 2 x^2 + 4 x+ 1 = 0

SECTION 9.2 The Quadratic Formula 511