FInd 1/(a - 1) + 1/(b - 1) if a and b are the roots of the equation 2x^2 - 7x + 2 = 11x - 15.
2x^2 -7x+2 = 11x -15
2x^2-18x+17 = 0 Quadratic formula shows roots = 7.92783 and 1.07217
plugging into 1/(a - 1) + 1/(b - 1)
= ~ 14
Using the work of Electric Pavlov:
If you leave the solutions in radical form, the answer is exactly 14.
Solutions: [ 18 + sqrt(188) ] / 4 ---> [ 18 + 2·sqrt(47) ] 4 ---> [ 9 + sqrt(47) ] / 2
and [ 18 + sqrt(188) ] / 4 ---> [ 18 - 2·sqrt(47) ] 4 ---> [ 9 - sqrt(47) ] / 2
If a = [ 9 + sqrt(47) ] / 2 then a - 1 = [ 7 + sqrt(47) ] / 2 and 1 / (a - 1) = 2 / [ 7 + sqrt(47) ]
If b = [ 9 - sqrt(47) ] / 2 then b - 1 = [ 7 - sqrt(47) ] / 2 and 1 / (b - 1) = 2 / [ 7 - sqrt(47) ]
Multiply both of these fractions by the complement of their divisors to get:
2[ 7 - sqrt(47) ] / [ 49 - 47 ] + 2[ 7 + sqrt(47) ] / [ 49 - 47 ]
= [ 14 - 2·sqrt(47) ] + [ 14 - 2·sqrt(47) ] / 2
= 28 / 2
= 14