Answer
$\left\{ \dfrac{3-\sqrt{57}}{4},\dfrac{3+\sqrt{57}}{4} \right\}$
Work Step by Step
Multiplying both sides of the given equation, $
\dfrac{3}{x}+\dfrac{4}{x+2}=2
,$ by the $LCD=
x(x+2)
$, then,
\begin{array}{l}\require{cancel}
x(x+2)\left( \dfrac{3}{x}+\dfrac{4}{x+2} \right)=\left( 2\right)x(x+2)
\\\\
(x+2)(3)+x(4)=2x^2+4x
\\\\
3x+6+4x=2x^2+4x
\\\\
-2x^2+(3x+4x-4x)+6=0
\\\\
-2x^2+3x+6=0
.\end{array}
Using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solutions of the quadratic equation, $
-2x^2+3x+6=0
,$ are
\begin{array}{l}\require{cancel}
\dfrac{-(3)\pm\sqrt{(3)^2-4(-2)(6)}}{2(-2)}
\\\\=
\dfrac{-3\pm\sqrt{9+48}}{-4}
\\\\=
\dfrac{-3\pm\sqrt{57}}{-4}
\\\\=
\dfrac{3\pm\sqrt{57}}{4}
.\end{array}
Upon checking, both solutions satisfy the original equation. Hence, the solutions are $
\left\{ \dfrac{3-\sqrt{57}}{4},\dfrac{3+\sqrt{57}}{4} \right\}
.$
