Question

If $\alpha$ and $\beta$ $(\alpha <\beta )$, are the roots of the equation $x^2+bx+c=0$, where $c<0<b$, then

• A. $0<\alpha <\beta$
• B. $\alpha <0<\beta <\left|\alpha \right|$
• C. $\alpha <\beta <0$
• D. $\alpha <0<\left|\alpha \right|<\beta$

Answer 1

The correct option is B $\alpha <0<\beta <\left|\alpha \right|$

$\alpha +\beta =-b$
And $b>0$ Therefore,
$\alpha +\beta <0$
And
$\alpha \beta =c$ and $c<0$
Therefore,
$\alpha \beta <0$
Now,
$\alpha <\beta$
Therefore, $\beta$ is positive root while $\alpha$ is negative root and $|\alpha |>|\beta |$ (since the sum of the roots is negative).
Hence,
$\alpha <0$ and $\beta >0$
Therefore, $\alpha <0<\beta <|\alpha |$.