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

Answer: 12

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 $\alpha +\beta +{\gamma }_1=-5$
$\alpha +\beta +{\gamma }_2=-7$
$\Rightarrow {\gamma }_1+{\gamma }_2=-12$