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

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 negative, 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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