First slide

Theory of equations

Question

If one root of the equation $x^{2} + px + q = 0$ is the square of the other, then

Moderate

A

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

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

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

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

Solution

Let root of the given equation $x^{2} + px + q = 0 are α$ and $α^{2}$
Now, $α α^{2} = α^{3} = q, α + α^{2} = - p$
Cubing both sides, $α^{3} + {(α^{2})}^{3} + 3 α . α^{2} (α + α^{2}) = - p^{3}$
$q + q^{2} + 3 q (- p) = - p^{3}$
$p^{3} + q^{2} + q (1 - 3 p) = 0$

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