Question

Number of real roots of the equation $\begin{vmatrix}1 & x & x \\ x & 1 & x \\ x & x & 1\end{vmatrix}+\begin{vmatrix}1-x & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x\end{vmatrix}=0$ is

Answer 1

$R_{1} \rightarrow R_{1}+R_{2}+R_{3}$
$\begin{vmatrix}2 x+1 & 2 x+1 & 2 x+1 \\ x & 1 & x \\ x & x & 1\end{vmatrix}+\begin{vmatrix}3-x & 3-x & 3-x \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x\end{vmatrix}=0$
$(2 x+1)\begin{vmatrix}1 & 1 & 1 \\ x & 1 & x \\ x & x & 1\end{vmatrix}+(3-x)\begin{vmatrix}1 & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x\end{vmatrix}=0$
$C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1}$
$(2 x+1)\begin{vmatrix}1 & 0 & 0 \\ x & 1-x & 0 \\ x & 0 & 1-x\end{vmatrix}+(3-x)\begin{vmatrix}1 & 0 & 0 \\ 1 & -x & 0 \\ 1 & 0 & -x\end{vmatrix}=0$