Question

Find the roots of equation $\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& 5\\ 1& 2x& 5x^2\end{array}\right|=0$.

Answer 1

$\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& 5\\ 1& 2x& 5x^2\end{array}\right|=0$

$\Rightarrow 1(-10x^2-10x)-4(5x^2-5)+20(2x+2)=0$

$\Rightarrow -10x^2-10x-20x^2+20+40x+40=0$

$\Rightarrow -30x^2+30x+60=0$

$\Rightarrow -30(x^2-x-2)=0$

$\Rightarrow x^2-x-2=0$

$\Rightarrow x=\frac{1\pm \sqrt{1+8}}{2}=\frac{1+3}{2}$

$\therefore x=2$ and $x=-1$