Solve for $x$:
$\frac{5}{x^2-9}+\frac{1}{6}=\frac{1}{x-3}$

$x=3$

$x=-3$

$x=3+\sqrt{6}$ only

$x=3-\sqrt{6}$ only

$x=3-\sqrt{6}$

$x=3+\sqrt{6}$