Question

$x=-9$ is root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$. The other two roots are $a,b$, then find $|b-a|?$

Answer 1

$\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
$\Rightarrow x(x^2-12)-3(2x-14)+7(12-7x)=0$
$\Rightarrow x^3-67x+126=0$
Its given $x=-9$ is root of $x^3-67x+126=0$
$\to x^3-67x+126=(x+9)(x^2-9x+14)$
$\Rightarrow$ other two roots $a,b$ are $2,7$
$\therefore |b-a|=5$