If alpha and beta are the roots of the equation x^2 - x - 3 = 0

then state the values of alpha + beta and hence find the values of
(alpha/(alpha + beta)/(alpha + beta)/beta

The standard result is that alpha + beta = -b/a, and alpha * beta = c/a, whera a, b & c are the coefficients of the terms of the quadratic ; so alpha + beta = 1/1 = 1; for the second one, we need to do a bit of manipulation first. Reminder - when you divide by a fraction, you invert the divisor and multiply instead so the expression becomes alpha/(alpha + beta) * beta/(alpha + beta) and this becomes (alpha*beta)/(alpha + beta)^2 = -3/1^2 = -3/1 = -3

x^2 - x - 3 = 0,

x = [1 +/- sqrt(1 +4*3)] / 2 = (1 +/- √13)/2,
Thus,
α = (1+√13)/2, AND, β = (1-√13)/2 >==================< ANSWER

α + β = 1/2 + 1/2 = 1 >========================< ANSWER

(α/(α + β)) / ((α + β) / β) = α β / (α + β)^2 =(1 -13) = -12 >===========< ANSWER