To prove the zero-factor property, we first assume that (If adoes equal 0,

then the property is proved already.) If then exists, and both sides of

can be multiplied by.

Thus, if aZ 0,then b=0,and the property is proved.

b= 0

1

a

#ab=^1

a

0

1

a

aZ0,^1 a ab= 0

aZ0.

344 CHAPTER 6 Factoring

CAUTION If then or However, if for example, it

is not necessarily true that or In fact, it is very likely that neither

norb= 6.The zero-factor property works only for a product equal to0.

a= 6 b= 6. a= 6

ab=0, a= 0 b=0. ab=6,

Using the Zero-Factor Property to Solve an Equation

Solve

Here, the product of and is 0. By the zero-factor property, the fol-

lowing must hold true.

or Zero-factor property

or Solve each of these equations.

x=

3

2

x=- 6 2 x= 3

x+ 6 = 0 2 x- 3 = 0

x+ 6 2 x- 3

1 x+ 6212 x- 32 = 0.

NOW TRY EXAMPLE 1

EXERCISE 1
Solve.

1 x+ 5214 x- 72 = 0

NOW TRY ANSWER

1. E-5,^74 F

CHECK

Let

0 = 0 ✓ True

01 - 152  0

x=6.

1 - 6 + 62321 - 62 - 34  0

1 x+ 6212 x- 32 = 0

Let

0 = 0 ✓ True

15

2

102  0

x=^32.

a

3

2

• 6 ba 2 #

3

2

- 3 b  0

1 x+ 6212 x- 32 = 0

Both solutions check, so the solution set is E-6,^32 F. NOW TRY

Since the product equals the equation of

Example 1has a term with a squared variable and is an example of a quadratic equa-

tion.A quadratic equation has degree 2.

1 x+ 6212 x- 32 2 x^2 + 9 x-18,

Quadratic Equation

An equation that can be written in the form

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

form is called standard form.

aZ 0,

ax^2 bxc0,