Question

If one root of `x^(2) - x - k = 0` is square of the other, then k =
A. `2 pm sqrt5`
B. `2 pm sqrt3`
C. `3 pm sqrt2`
D. `5 pm sqrt2`

Answer 1

Correct Answer - 1
Let `alpha and alpha^(2)` be the roots of `x^(2) - x - k = 0`. Then,
`alpha + alpha^(2) = 1 and alpha^(3) = -k`
or `(-k)^(1//3) + (-k)^(2//3) = 1`
or `-k^(1//3) + k^(2//3) = 1`
or `(k^(2//3) - k^(1//3))^(3) = 1`
or `k^(2) - k - 3k (k^(2//3) - K^(1//3)) = 1`
or `k^(2) - k - 3k (1) = 1`
or `k^(2) - 4k - 1 = 0`
or `k = 2 pm sqrt5`