Answer
$x=\dfrac{3}{4}\pm\dfrac{\sqrt{57}}{4}$
Work Step by Step
$\dfrac{3}{x}+\dfrac{4}{x+2}=2$
Multiply the whole equation by $x(x+2)$:
$x(x+2)\Big(\dfrac{3}{x}+\dfrac{4}{x+2}=2\Big)$
$3(x+2)+4x=2x(x+2)$
$3x+6+4x=2x^{2}+4x$
Take all terms to the right side of the equation and simplify it:
$0=2x^{2}+4x-3x-4x-6$
$2x^{2}-3x-6=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a=2$, $b=-3$ and $c=-6$.
Substitute the known values into the formula and simplify:
$x=\dfrac{-(-3)\pm\sqrt{(-3)^{2}-4(2)(-6)}}{2(2)}=\dfrac{3\pm\sqrt{9+48}}{4}=...$
$...=\dfrac{3\pm\sqrt{57}}{4}=\dfrac{3}{4}\pm\dfrac{\sqrt{57}}{4}$
The original equation is not undefined for neither of the values of $x$ found. Our final answer is $x=\dfrac{3}{4}\pm\dfrac{\sqrt{57}}{4}$
