Answer
$\left\{ \dfrac{9-\sqrt{105}}{2},\dfrac{9+\sqrt{105}}{2} \right\}$
Work Step by Step
Multiplying both sides of the given equation, $
\dfrac{5}{x-2}+\dfrac{4}{x+2}=1
,$ by the $LCD=
(x-2)(x+2)
$, then,
\begin{array}{l}\require{cancel}
(x-2)(x+2)\left( \dfrac{5}{x-2}+\dfrac{4}{x+2} \right)=\left( 1 \right)(x-2)(x+2)
\\\\
(x+2)(5)+(x-2)(4)=x^2-4
\\\\
5x+10+4x-8=x^2-4
\\\\
-x^2+(5x+4x)+(10-8+4)=0
\\\\
-x^2+9x+6=0
.\end{array}
Using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solutions of the quadratic equation, $
-x^2+9x+6=0
,$ are
\begin{array}{l}\require{cancel}
\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\pm\sqrt{105}}{2}
.\end{array}
Upon checking, both solutions satisfy the original equation. Hence, the solutions are $
\left\{ \dfrac{9-\sqrt{105}}{2},\dfrac{9+\sqrt{105}}{2} \right\}
.$
