Question

If $x^3+5x^2+px+q=0$ and $x^3+7x^2+px+r=0$ have two roots in common and their third roots are ${\gamma }_1$ and ${\gamma }_2$ respectively, then the value of $|{\gamma }_1+{\gamma }_2|$ is

Answer 1

Let the roots of $x^3+5x^2+px+q=0\cdots \left(i\right)$ be $\alpha ,\beta ,{\gamma }_1$
and roots of $x^3+7x^2+px+r=0\cdots \left(ii\right)$ be $\alpha ,\beta ,{\gamma }_2.$
Subtracting equation $\left(ii\right)$ from $\left(i\right),$
$-2x^2+q-r=0\cdots \left(iii\right)$
$\because \alpha ,\beta$ are common roots, so the roots of the equation $\left(iii\right)$ are $\alpha ,\beta .$
$\Rightarrow \alpha +\beta =0$

From equation $\left(i\right),$
$\alpha +\beta +{\gamma }_1=-5\Rightarrow {\gamma }_1=-5$
From equation $\left(ii\right),$
$\alpha +\beta +{\gamma }_2=-7\Rightarrow {\gamma }_2=-7$

$\therefore {\gamma }_1+{\gamma }_2=-12$
$\Rightarrow |{\gamma }_1+{\gamma }_2|=12$