$$\frac{1}{x-1}+2x = 0$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 2 }+\dfrac{ 1 }{ 2 }i & x_2 = \dfrac{ 1 }{ 2 }-\dfrac{ 1 }{ 2 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x-1}+2x &= 0&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)\cdot\frac{1}{x-1}+(x-1)\cdot2x &= (x-1)\cdot0&& \text{cancel out the denominators} \\[1 em]1+2x^2-2x &= 0&& \text{simplify left side} \\[1 em]2x^2-2x+1 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}-2x+1 = 0 $ is a quadratic equation.
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