If one of the roots of the equation $\left|\begin{array}{l}7\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\\ 6\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\\ x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}7\end{array}\right|=0$ is 7, then the other roots are

$\left|\begin{array}{l}7\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\\ 6\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\\ x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}7\end{array}\right|=\left|\begin{array}{l}9+x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\\ 9+x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\\ 9+x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}7\end{array}\right|=0$

$\begin{array}{r}\Rightarrow (9+x)\left|\begin{array}{lll}1& 2& x\\ 1& x& 3\\ 1& 2& 7\end{array}\right|=0\\ \Rightarrow x=-9\end{array}$

When X = 2, $c_1= c_2$

When x = 7, $R_1 = R_3$

Values of x are (2, 7, -9). 

$Aliter$

7 is a root (given) 

-9 is a root, by inspection $\left(C_1\to C_1+C_2+C_3\right)$

Products of roots 

$=\left|\begin{array}{l}7\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 6\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}7\end{array}\right|=9\left|\begin{array}{ccc}1& 2& 0\\ 0& -2& 3\\ 0& 0& 7\end{array}\right|=-\mathrm{9.2.7}$

$\therefore$the other root is 2 

So (2, -9)