First slide

Theory of equations

Question

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

$∵ α + β = - b, α β = c$ ... (i)
$∵ c < 0 \Rightarrow α β < 0$
Let $α < 0, β > 0$
$∴ | α | = - α$ and $α < 0 < β [∵ α < β]$ ... (ii)
From Eq. (i), we get $- | α | + β < 0$
$\Rightarrow β < | α |$ ... (iii)
From Eqs. (ii) and (iii), we get
$α < 0 < β < | α |$

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