Beginning Algebra, 11th Edition

OBJECTIVE 2 Solve equations of the form where In

Example 1(b),we might also have solved by noticing that xmust be a num-

ber whose square is 9. Thus, or This is generalized

as the square root property.

x= 29 = 3 x=- 29 =-3.

x^2 = 9

x^2 k, k>0.

554 CHAPTER 9 Quadratic Equations

OBJECTIVES In Section 6.5,we solved quadratic equations by factoring. Since not all quadratic

equations can easily be solved by factoring, we must develop other methods.

OBJECTIVE 1 Review the zero-factor property. Recall that a quadratic

equationis an equation that can be written in the form

Standard form

for real numbers a, b, and c, with As seen in Section 6.5,we can solve the

quadratic equation x^2 + 4 x+ 3 = 0 by factoring, using the zero-factor property.

aZ0.

ax^2 bxc0,

Solving Quadratic Equations by the Square Root Property

9.1

1 Review the zero-

factor property.

2 Solve equations of

the form

where

3 Solve equations

of the form

where

4 Use formulas

involving squared

variables.

k 7 0.

1 ax+b 22 =k,

k 7 0.

x 2 =k,

Zero-Factor Property

If aand bare real numbers and if ab=0,then a= 0 or b=0.

Solving Quadratic Equations by the Zero-Factor Property

Solve each equation by the zero-factor property.

(a)

Factor.

or Zero-factor property

or Solve each equation.

The solution set is

(b)

Subtract 9.

Factor.

or Zero-factor property

or Solve each equation.

The solution set is 5 - 3, 3 6. NOW TRY

x=- 3 x= 3

x+ 3 = 0 x - 3 = 0

1 x+ 321 x- 32 = 0

x^2 - 9 = 0

x^2 = 9

5 - 3, - 16.

x=- 3 x=- 1

x+ 3 = 0 x + 1 = 0

1 x+ 321 x+ 12 = 0

x^2 + 4 x+ 3 = 0

EXAMPLE 1

Square Root Property

If kis a positive number and if then

or

The solution set is which can be written ( is read

“positive or negative” or “plus or minus.”)

E 2 k, 2 kF, E 2 kF. 

x 2 k x 2 k.

x^2 =k,

NOW TRY
EXERCISE 1
Solve each equation by the
zero-factor property.

(a)

(b)x^2 = 36

x^2 - x- 20 = 0

NOW TRY ANSWERS

1. (a) 5 - 4, 5 6 (b) 5 - 6, 6 6

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