Question

Given that $x=-9$ is a root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$. The sum of other two roots is

Answer 1

Given $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
Applying $R_1\to R_1+R_2+R_3$

$\Rightarrow \left|\begin{array}{ccc}x+9& x+9& x+9\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
$\Rightarrow (x+9)\left|\begin{array}{ccc}1& 1& 1\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$

Applying $C_2\to C_2-C_1,C_3\to C_3-C_1$
$\Rightarrow (x+9)\left|\begin{array}{ccc}1& 0& 0\\ 2& x-2& 0\\ 7& -1& x-7\end{array}\right|=0\Rightarrow (x+9)(x-2)(x-7)=0$
$\Rightarrow x=-9,2,7$ are the roots
Therefore sum of other roots is $2+7=9$