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

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

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

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

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

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Solution

The correct option is A
$p^{3} - q (3 p - 1) + q^{2} = 0$

Let the roots of $x^{2} + p x + q = 0$ be $α$ and $α^{2} .$
$\Rightarrow α + α^{2} = - p a n d α^{3} = q \Rightarrow α (α + 1) = - p \Rightarrow α^{3} {α^{3} + 1 + 3 α (α + 1)} = - p^{3}$ [cubing both sides]
$\Rightarrow q (q + 1 - 3 p) = - p^{3} \Rightarrow p^{3} - (3 p - 1) q + q^{2} = 0$

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