Answer
$\displaystyle x=\frac{9\pm\sqrt{73}}{2}$
Work Step by Step
$\displaystyle \frac{3x}{x-2}+\frac{1}{x}=4$
We multiply through by $x(x-2)$:
$\displaystyle x(x-2)(\frac{3x}{x-2}+\frac{1}{x})=x(x-2)*4$
$3x*x+(x-2)=4x^{2}-8x$
$3x^{2}+x-2=4x^{2}-8x$
$x^{2}-9x+2=0$
We solve using the quadratic formula ($a=1,\ b=-9,\ c=2$):
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$\displaystyle x=\frac{-(-9)\pm\sqrt{(-9)^{2}-4*1*2}}{2*1}$
$\displaystyle x=\frac{9\pm\sqrt{81-8}}{2}$
$\displaystyle x=\frac{9\pm\sqrt{73}}{2}$
