If one root is square of the other root of the equation `x^2+p x+q=0,` then the relation between `pa n dq` is `p^3-q(3p-1)+q^2=0` `p^3-q(3p+1)+q^2=0` `p^3+q(3p-1)+q^2=0` `p^3+q(3p+1)+q^2=0`

Options

A. `p^(3) - (3p - 1) q + q^(2) = 0`
B. `p^(3) - (3p + 1) q + q^(2) = 0`
C. `p^(3) + (3p - 1) q + q^(2) = 0`
D. `p^(3) + (3p + 1)q + q^(2) = 0`

Answer

A

Explanation

Let `alpha, alpha^(2)` be the roots of the equation `x^(2) + px + q = 0`. Then,
`alpha + alpha^(2) = - p and alpha xx alpha^(2) = q`
`rArr" "alpha + alpha^(2) = -p and alpha = q^(1//3)`
`rArr" "q^(1//3) + q^(2//3) = -p`
`q + q^(2) - 3pq = - p^(3) rArr p^(3) - q (3p -1) q^(2) = 0`