Intermediate Algebra (11th edition)

496 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

OBJECTIVES Recall from Section 6.5that a quadratic equationis defined as follows.

The Square Root Property and Completing the Square

9.1

1 Review the

zero-factor

property.

2 Learn the square

root property.

3 Solve quadratic

equations of the

form

by extending the

square root

property.

4 Solve quadratic

equations by

completing the

square.

5 Solve quadratic

equations with

solutions that are

not real numbers.

1 ax+b 22 =c

Quadratic Equation

An equation that can be written in the form

where a, b, and care real numbers, with is a quadratic equation.The

given form is called standard form.

aZ0,

ax^2 bxc0,

A quadratic equation is a second-degree equation,that is, an equation with a

squared variable term and no terms of greater degree.

4 x^2 + 4 x- 5 = 0 and 3 x^2 = 4 x- 8

Quadratic equations

( The first equation is

in standard form.)

OBJECTIVE 1 Review the zero-factor property.In Section 6.5we used fac-

toring and the zero-factor property to solve quadratic equations.

Zero-Factor Property

If two numbers have a product of 0, then at least one of the numbers must be 0.

That is, if ab= 0,then a= 0 or b=0.

Using the Zero-Factor Property

Solve by using the zero-factor property.

Factor.

or Zero-factor property

or Solve each equation.

To check, substitute each solution in the original equation. The solution set is E-

7

3 , 4F.

x=-

7

3

3 x=- 7 x= 4

3 x+ 7 = 0 x - 4 = 0

13 x+ 721 x- 42 = 0

3 x^2 - 5 x- 28 = 0

3 x^2 - 5 x- 28 = 0

EXAMPLE 1

Square Root Property

If xand kare complex numbers and then

x 2 k or x 2 k.

x^2 =k,

NOW TRY
EXERCISE 1
Use the zero-factor property
to solve 2x^2 + 5 x- 12 =0.

NOW TRY ANSWER

1. E-4,^32 F

NOW TRY

OBJECTIVE 2 Learn the square root property. Not every quadratic equation

can be solved easily by factoring. Three other methods of solving quadratic equations

are based on the following property.