Objective 14 – Null Factor Law and Solving Quadratic Equations:

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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

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