Question
$$\frac{2}{x}-2+\frac{1}{x}+1 = x\cdot2-x-2$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 13 }}{ 2 } & x_2 = \dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 13 }}{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{2}{x}-2+\frac{1}{x}+1 &= x\cdot2-x-2&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{2}{x}-x\cdot2+x\cdot\frac{1}{x}+x\cdot1 &= xx\cdot2-xx-x\cdot2&& \text{cancel out the denominators} \\[1 em]2-2x+1+x &= 2x^2-x^2-2x&& \text{simplify left and right hand side} \\[1 em]-x+3 &= x^2-2x&& \text{move all terms to the left hand side } \\[1 em]-x+3-x^2+2x &= 0&& \text{simplify left side} \\[1 em]-x^2+x+3 &= 0&& \\[1 em] \end{aligned} $$
$ -x^{2}+x+3 = 0 $ is a quadratic equation.
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