Question

Find the integral value of x, if $\left|\begin{array}{ccc}{x}^2& x& 1\\ 0& 2& 1\\ 3& 1& 4\end{array}\right|=28.$

Answer 1

$$\begin{gathered} \mathrm{Given}:\left|\begin{array}{ccc}{x}^2& x& 1\\ 0& 2& 1\\ 3& 1& 4\end{array}\right|=28 \\ \Rightarrow x^2\left(8-1\right)-x\left(0-3\right)+1\left(0-6\right) \\ \Rightarrow 8x^2-x^2+3x-6=28 \\ \Rightarrow 7x^2+3x-6=28 \\ \Rightarrow 7x^2+3x-34=0 \\ \Rightarrow \left(7x+17\right)\left(x-2\right)=0 \\ \Rightarrow x=2 \end{gathered}$$

Integral value of x is 2. Thus, $x=\frac{-17}{7}$ is not an integer.