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

 Sep 7, 2023
 #1
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Note that       1/ ( a - 1)  +  1/(b -1)    can  be  simplified to

 

[ (a + b)  - 2 ]   / ( ab  - (a + b) + 1 )       .......    (1)

 

Simplify the equation as

 

3x^2 - 11x  - 7  =  0

 

By Vieta   the sum of the  roots  (a + b)  =  11/3

 

And the  product of  the  roots  ab   = -7/3

 

Subbing into (1) we have

 

[ 11/3 - 2 ]  /  [ -7/3 - ( 11/3) + 1 ]  =

 

[  5/3 ] / [ -5 ] =

 

-5/15  =

 

-1/3

 

cool cool cool

 Sep 7, 2023

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