First slide

Theory of equations

Question

If $α and β (α < β)$ are the roots of the equation $x^{2} + bx + c = 0, where c < 0 < b, then$

Moderate

A

$0 < α < β$
B

$α < 0 < β < | α |$
C

$α < β < 0$
D

$α < 0 < | α | < β ∣$

Solution

Since $α, β$ are the roots of the equation $x^{2} + bx + c = 0 .$ Here, $D = b^{2} - 4 c > 0 because c < 0 < b .$ so, roots are real and unequal.
Now , $α + β = - b < 0 and αβ = c < 0$
$∴$ One root is positive and the other is negative, then the negative root being numerically bigger. As, $α < β, α$ is the negative root while $β$ is the positive root. so $| α | > β$
and $α < 0 < β$

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