Answer
$\displaystyle \frac{5\pm 3\sqrt{5}}{2}=\frac{5}{2}\pm\frac{3\sqrt{5}}{2}$
Work Step by Step
$\displaystyle \frac{x}{x+3}=\frac{2}{x-3}-\frac{1}{x^{2}-9}$
We multiply through by $(x-3)(x+3)$:
$x(x-3)=2(x+3)-1$
And distribute:
$x^{2}-3x=2x+6-1$
$x^{2}-3x-2x-6+1=0$
$x^{2}-5x-5=0$
Now we use the quadratic formula $(a=1, b=-5, c=-5)$:
$x=\displaystyle \frac{-(-5)\pm\sqrt{(-5)^{2}-4*1*-5}}{2}=\frac{5\pm 3\sqrt{5}}{2}=\frac{5}{2}\pm\frac{3\sqrt{5}}{2}$
