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Discriminant

If α and ...

Question

If $α$ and $β$ be two real roots of the equation $x^{3} + p x^{2} + q x + r = 0$ satisfying the relation $α β + 1 = 0$ , then prove that $r^{2} + p r + q + 1 = 0$ .

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Solution

$α β + 1 = 0 α β = - 1$
$α β γ = - r$
$(+ 1) γ = - r$
$γ = r$
$∴ r^{3} + p r^{2} + q r + r = 0$
$r (r^{2} + p r + q + 1) = 0$
$∴ r^{2} + p r + q + 1 = 0$
Hence, proved.

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