Using properties of determinants, solve for `x:|a+x a-x a-x a-x a+x a-x a-x a-x a+x|=0`

Answer

`x=0, x=3 a`

Explanation

$$ Given, \left|\begin{array}{lll}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0$$
$$ \text{Applying} R_{2} \rightarrow R_{2}-R_{1} \text{ and } R_{3} \rightarrow R_{3}-R_{1}$$
$$
\begin{aligned}
&\Rightarrow\left|\begin{array}{ccc}
a+x & a-x & a-x \\
-2 x & 2 x & 0 \\
-2 x & 0 & 2 x
...