Question

Find $x$ so that
$\left[\begin{array}{ccc}1& x& 1\end{array}\right]\left[\begin{array}{ccc}1& 3& 2\\ 0& 5& 1\\ 0& 3& 2\end{array}\right]\left[\begin{array}{c}1\\ 1\\ x\end{array}\right]=0$
if $x=\frac{-9\pm \sqrt{k}}{2}$
what is $k?$

Answer 1

We have,
$\left[\begin{array}{ccc}1& x& 1\end{array}\right]\left[\begin{array}{ccc}1& 3& 2\\ 0& 5& 1\\ 0& 3& 2\end{array}\right]\left[\begin{array}{c}1\\ 1\\ x\end{array}\right]=0$
$\Rightarrow \left[\begin{array}{ccc}1& 5x+6& x+4\end{array}\right]\left[\begin{array}{c}1\\ 1\\ x\end{array}\right]=0$
$\Rightarrow \left[\begin{array}{c}1+5x+6+x^2+4x\end{array}\right]=0$
$or{x}^2+9x+7=0$
$\therefore x=\frac{-9\pm \sqrt{(81-28)}}{2}$
$\therefore x=\frac{-9\pm \sqrt{53}}{2}$

$\therefore k=53$