Answer
$x=\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$
Work Step by Step
$\dfrac{5}{x-2}+\dfrac{4}{x+2}=1$
Multiply the whole equation by $(x-2)(x+2)$:
$(x-2)(x+2)\Big(\dfrac{5}{x-2}+\dfrac{4}{x+2}=1\Big)$
$5(x+2)+4(x-2)=(x-2)(x+2)$
$5x+10+4x-8=x^{2}-4$
Take all terms to the right side of the equation and simplify it:
$0=x^{2}-4-5x-10-4x+8$
$x^{2}-9x-6=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a=1$, $b=-9$ and $c=-6$.
Substitute the known values into the formula and simplify:
$x=\dfrac{-(-9)\pm\sqrt{(-9)^{2}-4(1)(-6)}}{2(1)}=\dfrac{9\pm\sqrt{81+24}}{2}=...$
$...=\dfrac{9\pm\sqrt{105}}{2}=\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$
The original equation is not undefined for neither of the values of $x$ found. Our final answer is $x=\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$
