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Discriminant

If α, β be th...

Question

If $α, β$ be the real roots of the equation $x^{2} + a x + 1 = 0$ .Then the minimum value of $α^{2} + β^{2}$ is

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Solution

The correct option is D $2$
$α + β = - a, α \times β = 1$
$α^{2} + β^{2} = (α + β)^{2} - 2 α β α^{2} + β^{2} = a^{2} - 2$
As the roots are real.
So, $a^{2} - 4 \geq 0$
$a^{2} \geq 4$
So minimum value of $α^{2} + β^{2} = 4 - 2 = 2$

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