Q.Solve : $\begin{vmatrix} 7 & 4 & 11 \\ -3 & 5 & x \\ -x & 3 & 1 \end{vmatrix}=0$

Expand the determinant along the first row, simplify to $2x^2-17x+26=0$, and solve: $x=2$ or $x=\tfrac{13}{2}$.

Given: $\begin{vmatrix} 7 & 4 & 11 \\ -3 & 5 & x \\ -x & 3 & 1 \end{vmatrix}=0.$

Step 1 — expand along the first row.

$7\begin{vmatrix}5 & x\\3 & 1\end{vmatrix}-4\begin{vmatrix}-3 & x\\-x & 1\end{vmatrix}+11\begin{vmatrix}-3 & 5\\-x & 3\end{vmatrix}=0.$

Step 2 — evaluate each $2\times2$ minor.

$7(5-3x)-4(-3+x^2)+11(-9+5x)=0.$

Step 3 — simplify.

$35-21x+12-4x^2-99+55x=0\;\Rightarrow\;-4x^2+34x-52=0.$

Divide by $-2$: …