Intermediate Algebra (11th edition)

Using the Quadratic Formula (Nonreal Complex Solutions)

Solve

Multiply.

Standard form Add 8.

From the equation we identify and.

Quadratic formula

Substitute.

Simplify.

Factor.

Lowest terms

Standard form for a

complex number

The solution set is NOW TRY

OBJECTIVE 3 Use the discriminant to determine the number and type of

solutions.The solutions of the quadratic equation are given by

Discriminant

If a, b, and care integers, the type of solutions of a quadratic equation—that is, rational,

irrational, or nonreal complex — is determined by the expression under the radical sym-

bol, called the discriminant( because it distinguishes among the three types

of solutions). By calculating the discriminant, we can predict the number and type of

solutions of a quadratic equation.

b^2 - 4 ac,

x=

- b 2 b^2 - 4 ac

2 a

.

ax^2 +bx+ c= 0

E

1

3 

2

3 iF.

x=^1 a+bi

3



2

3

i

x=

1  2 i

3

x=

611  2 i 2

6132

x= 2 - 144 = 12 i

6  12 i

18

x=

6  2 - 144

18

x=

- 1 - 62  21 - 622 - 4192152

2192

x=

- b 2 b^2 - 4 ac

2 a

9 x^2 - 6 x+ 5 =0, a=9, b=-6, c= 5

9 x^2 - 6 x+ 5 = 0

9 x^2 - 6 x- 3 =- 8

19 x+ 321 x- 12 =- 8

19 x+ 321 x- 12 =-8.

EXAMPLE 3

508 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

NOW TRY
EXERCISE 3
Solve. 1 x+ 521 x- 12 =- 18

NOW TRY ANSWER

3. 5 - 2  3 i 6

Discriminant

The discriminantof is If a, b, and care integers,

then the number and type of solutions are determined as follows.

ax^2 +bx+ c= 0 b^2  4 ac.

Number and

Discriminant Type of Solutions

Positive, and the square of an integer Two rational solutions

Positive, but not the square of an integer Two irrational solutions

Zero One rational solution

Negative Two nonreal complex solutions