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Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$  Find $\frac{1}{a-1}+\frac{1}{b-1}.$

 Aug 25, 2023
 #1
avatar+129690 
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1/ (a - 1)  +  1/ (b - 1)  =     (a + b - 2)  / ( ab - (a + b)  + 1)

 

Rearrange the equation  as

3x^2 -11x  - 7  =  0

The sum of the  roots =  a + b =  11/3

The product of the  roots =   ab  =  (-7) / 3

 

Then  we have   1/ (a - 1)  + 1 /(b - 1)  =   [ 11/3 - 2 ] / [ -7/3 - 11/3 + 1] =

 

[ 11/3 - 6/3 ]  /  [ -18/3 + 1 ]  =

 

[ 5/3 ]  /  [ -6 + 1 ]  =

 

(5/3) / (-5)  = 

 

5 / -15  =

 

-1/3

 

cool cool cool

 Aug 25, 2023

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