Question

If $x,y,z\in R,\Delta =\left|\begin{array}{ccc}x& x+y& x+y+z\\ 2x& 5x+2y& 7x+5y+2z\\ 3x& 7x+3y& 9x+7y+3z\end{array}\right|=-16$, then the real value of $x$ is

Answer 1

$\left|\begin{array}{ccc}x& x+y& x+y+z\\ y& 5x+2y& 7x+5y+2z\\ z& 7x+3y& 9x+7y+3z\end{array}\right|=-16=\left|\begin{array}{ccc}x& x& x\\ 2x& 5x& 7x\\ 3x& 7x& 9x\end{array}\right|+\left|\begin{array}{ccc}0& y& y\\ 0& 2y& 5y\\ 0& 3y& 7y\end{array}\right|+\left|\begin{array}{ccc}0& 0& z\\ 0& 0& 2z\\ 0& 0& 3z\end{array}\right|=\left|\begin{array}{ccc}x& x& x\\ 2x& 5x& 7x\\ 3x& 7x& 9x\end{array}\right|+0+0=-16=x[{45x}^2-{49x}^2]-x[{18x}^2-{21x}^2]-1x[{14x}^2-{15x}^2]=-16\Rightarrow x[-{4x}^2]-x\left[{-3x}^2\right]+x\left[{-x}^2\right]=-16\Rightarrow -{4x}^3{+3x}^3{-x}^3=-16\Rightarrow {-x}^3{-x}^3=-16\Rightarrow {-2x}^3=-16\Rightarrow x^3=8\Rightarrow x=2$