+501 Find $\frac{1}{a-1}+\frac{1}{b-1},$ where $a$ and $b$ are the roots of the quadratic equation $2x^2-7x+2 = 0.$
+128408 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
