Let a and b be the roots of 2x^2 + 4x - 5 = 0. Find
(a(1 - b) + b)/(a^2 + b^2).
I tried finding a and b, but that didn't help.
vieta's formulas would really help in this situation! With vietas, you don't need to find the specific roots a,b of the quadratic, but rather, you can derive and find the elementary symmetric sums of the roots (a + b + c, ab + bc + ac, abc as an example for a third degree poly.)
By vieta's, the roots a,b of the given quadratic are such that:
a+b = -4/2 = -2
ab = -5/2
Now that we have these two values, just expand the numerator as:
((a+b)-ab) = (-2+5/2) = 1/2 for the numerator
and rewrite the denominator as:
(a+b)^2 - 2ab = (-2)^2 +5 = 4 + 5 = 9
our answer is then:
(1/2)/9 = 1/18