Answer
$x=3\pm\sqrt{7}$
Work Step by Step
$\dfrac{2}{x}+\dfrac{3}{x-1}=1$
Multiply the whole equation by $x(x-1)$
$x(x-1)\Big(\dfrac{2}{x}+\dfrac{3}{x-1}=1\Big)$
$2(x-1)+3x=x(x-1)$
$2x-2+3x=x^{2}-x$
Take all terms to the right side of the equation and simplify it by combining like terms:
$0=x^{2}-x-2x-3x+2$
$x^{2}-6x+2=0$
Solve this equation using the quadratic formula, which is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. In this case, $a=1$, $b=-6$ and $c=2$.
Substitute the known values into the formula and simplify:
$x=\dfrac{-(-6)\pm\sqrt{(-6)^{2}-4(1)(2)}}{2(1)}=\dfrac{6\pm\sqrt{36-8}}{2}=...$
$...=\dfrac{6\pm\sqrt{28}}{2}=\dfrac{6\pm2\sqrt{7}}{2}=3\pm\sqrt{7}$
The original equation is not undefined for neither of the values of $x$ found. The final answer is $x=3\pm\sqrt{7}$
