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Objective 14 – Null Factor Law and Solving Quadratic Equations:

What is the null factor law?

Given that

Then either

:

*AB*

*A*

0

0 or

*B*

0

Why is this important to know?

We will use the null factor law to solve quadratic equations.

Ex: Solve for x:

*x*

2

*x*

72

0

In this problem it is hard to isolate x. However if we factor the left hand side of the equation we create a situation where we are multiplying two things together that equals zero:

*x*

2

*x*

72

(

*x*

9 )(

*x*

8 )

0

0

Now we will apply the null factor law: (

means therefore

*x*

2

*x*

72

0

(

*x*

either

9 )(

*x*

(

*x*

9 )

8 )

0

0 or (

*x*

8 )

0

)

To finish the problem we must solve for x in each equation:

*x*

2

*x*

72

(

*x*

9 )(

*x*

8 )

0

0

either (

*x*

9 )

0 or (

*x*

8 )

0

*x*

9 or

*x*

8

What do the answers

*x x*

9 and

*x*

8 mean to the problem?

9 and

*x*

8 are the values of

*x*

that make the equation true let

*x*

9

*x*

2

*x*

72

0 let

*x*

8

*x*

2

*x*

72

0

(

9 )

2

(

9 )

81

81

0

72

0

0

0 This is true and

( 8 )

2

( 8 )

72

72

72

0

0

0

0 This is true

More Examples:

*x*

2

*x*

0

*x*

(

*x*

a)

either

*x*

1 )

0 or

*x*

0

1

0

*x*

0 or

*x*

1

9

*x*

3

*x*

0

(

*x*

)( 9

*x*

2

1 )

0 b) (

*x*

)( 3

*x*

either

*x*

1 )( 3

*x*

1 )

0 or 3

*x*

1

0

0 or 3

*x*

1

0

*x*

0 or

*x*

1

3 or

*x*

1

3

*x*

2

12

*x*

11

0

( c)

either

*x x*

11

11 )(

*x*

1 )

0 or

*x*

0

1

*x*

-11 or

*x*

1

0

2

*x*

2

*x*

6

0

2

*x*

2

4

*x*

3

*x*

6

0 d)

( 2

*x*

)(

*x*

(

*x*

2

2 )( 2

*x*

)

3

3 (

*x*

)

2 )

0

0

either

*x*

2

0 or 2

*x*

3

0

*x*

-2 or

*x*

3

2