$$\frac{2}{x}-3+\frac{x}{2} = \frac{1}{2}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 7 }{ 2 }-\dfrac{\sqrt{ 33 }}{ 2 } & x_2 = \dfrac{ 7 }{ 2 }+\dfrac{\sqrt{ 33 }}{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{2}{x}-3+\frac{x}{2} &= \frac{1}{2}&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2\cdot\frac{2}{x}-x\cdot2\cdot3+x\cdot2\frac{x}{2} &= x\cdot2\cdot\frac{1}{2}&& \text{cancel out the denominators} \\[1 em]4-6x+x^2 &= x&& \text{simplify left side} \\[1 em]x^2-6x+4 &= x&& \text{move all terms to the left hand side } \\[1 em]x^2-6x+4-x &= 0&& \text{simplify left side} \\[1 em]x^2-7x+4 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}-7x+4 = 0 $ is a quadratic equation.
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