+1348 Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a-1}+\frac{1}{b-1}.$
+129771 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
