Question

The solution to the equation:

$\left|\begin{array}{ccc}x& 2& -1\\ 2& 5& x\\ -1& 2& x\end{array}\right|=0$

Answer 1

We have,

$\left|\begin{array}{ccc}x& 2& -1\\ 2& 5& x\\ -1& 2& x\end{array}\right|=0$

Now solving the value of the determinant.

$\begin{array}{rcl}x(5x-2x)-2(2x+x)-(4+5)& =& 0\\ 5x^2-2x^2-4x-2x-9& =& 0\\ 3x^2-6x-9& =& 0\\ x^2-2x-3& =& 0\\ (x+1)(x+3)& =& 0\end{array}$

From this, we can easily compute the value of $x$ as $-1$ and $3$.

Hence, the solution is $-1$,$3$.