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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