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Find $\frac{1}{a-1}+\frac{1}{b-1},$ where $a$ and $b$ are the roots of the quadratic equation $2x^2-7x+2 = 0.$

 Apr 16, 2021
 #1
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By Vieta....

 

sum  of  the roots   a + b  =  - (-7)/ 2 =  7/2

 

product of the roots   ab   =  2/2  = 1

 

Note  that     

 

1/ (a - 1)  + 1/ (b - 1)  =    ( b - 1 + a - 1) / ( ab - a - b + 1)  =   ( a +b - 2)  / ( ab + 1 - ( a + b))

 

Filling  in what  we  know  we have

 

( 7/2 - 2)  / ( 1 + 1 - (7/2) )  =

 

( 3/2)  / ( -3/2)  = 

 

-1

 

cool cool cool

 Apr 16, 2021

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