Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a-1}+\frac{1}{b-1}.$
+128707 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
