Question

$x^{3}+5x^{2}+px+q=0$ and $x^{3}+7x^{2}+px+r=0,$ have two roots in common. If their third roots are $\gamma _{1}$ and $\gamma _{2}$ respectively, then $\left|\gamma _{1} + \gamma _{2}\right|$ is equal to

Answer 1

Let $\alpha \&\beta $ are common roots
$\therefore \alpha^{3}+5 \alpha^{2}+p \alpha+q=0$
$\alpha^{3}+7 \alpha^{2}+p \alpha+r=0$
$2\alpha ^{2}+r-q=0$ has roots $\alpha \&\beta $
$\Rightarrow \alpha +\beta =0$
since $\alpha +\beta +\gamma _{1}=-5$
$ \, \alpha +\beta +\gamma _{2}=-7$
$\Rightarrow \gamma _{1}+\gamma _{2}=-12$