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Let A and B be real numbers such that A/(x - 5) + B/(x + 1) = (12x + 22)/(x^2 - 4x - 5). What is A+B?

 Aug 2, 2022
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Note that  x^2  - 4x - 5 =  ( x - 5) ( x + 1)

 

Multiply through by these two factors and we  get

 

12x + 22  =   A( x + 1)  + B ( x -5)       simplify

 

12x + 22 =  (A + B)x  +  ( A - 5B)        equating terms we have

 

A + B  = 12     →   B = 12 -  A     (1)

 

A - 5B  =  22       (2)

 

Sub (1) into (2)

 

A - 5 ( 12 - A) = 22

 

A - 60 + 5A  = 22

 

6A  = 82

 

A =  82/6  =  41/3

 

And   

 

B   = 12  - 41/3  =   36/3 - 41/3   =  - 5 / 3

 

A + B  =   36/3  =  12

 

cool cool cool

 Aug 2, 2022
edited by CPhill  Aug 2, 2022

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