If one root is square of the other root of the equation x 2 + p x + q = 0 , then the relation between p and q is [2004]

Question

If one root is square of the other root of the equation $x^{2} + p x + q = 0,$ then the relation between $p$ and $q$ is [2004]

Smart Achievers · Accepted Answer

(1)

$x^{2} + p x + q = 0$

$Let roots be α and α^{2}, then$

$α + α^{2} = - p, α \cdot α^{2} = q \Rightarrow α = q^{1 / 3}$

$∴$ ${(q)}^{1 / 3} + {(q^{1 / 3})}^{2} = - p$

$On taking cube on both sides, we get$

$q + q^{2} + 3 q (q^{1 / 3} + q^{2 / 3}) = - p^{3}$

$\Rightarrow q + q^{2} - 3 p q = - p^{3}$

$\Rightarrow p^{3} + q^{2} - q (3 p - 1) = 0$

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