Question
$$\frac{x-2}{x}+\frac{11}{2}x = \frac{3}{2}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 22 }-\dfrac{\sqrt{ 177 }}{ 22 } & x_2 = \dfrac{ 1 }{ 22 }+\dfrac{\sqrt{ 177 }}{ 22 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x-2}{x}+\frac{11}{2}x &= \frac{3}{2}&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2 \cdot \frac{x-2}{x}+x\cdot2\frac{11}{2}x &= x\cdot2\cdot\frac{3}{2}&& \text{cancel out the denominators} \\[1 em]2x-4+11x^2 &= 3x&& \text{simplify left side} \\[1 em]11x^2+2x-4 &= 3x&& \text{move all terms to the left hand side } \\[1 em]11x^2+2x-4-3x &= 0&& \text{simplify left side} \\[1 em]11x^2-x-4 &= 0&& \\[1 em] \end{aligned} $$
$ 11x^{2}-x-4 = 0 $ is a quadratic equation.
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