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

$\because$         $\alpha +\beta =-b,\alpha \beta =c$                           …..(i)

$\because$                   $c<0\Rightarrow \alpha \beta <0$

Let         $\alpha <0,\beta >0$

      $\left(\therefore \left|\alpha \right|=-\alpha \right)$                           $\left(\because \alpha <\beta \right)$

$\Rightarrow$        $\alpha <0<\beta$                            ……..(ii)

From Eq. (i), $-\left|\alpha \right|+\beta <0\Rightarrow \beta <\left|\alpha \right|$        ……..(iii)

From Eq. (ii) and (iii), we get    $\alpha <0<\beta <\left|\alpha \right|$ .