$$\frac{4}{x} = \frac{9}{2}-x$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 9 }{ 4 }-\dfrac{\sqrt{ 17 }}{ 4 } & x_2 = \dfrac{ 9 }{ 4 }+\dfrac{\sqrt{ 17 }}{ 4 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{4}{x} &= \frac{9}{2}-x&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2\cdot\frac{4}{x} &= x\cdot2\cdot\frac{9}{2}-x\cdot2x&& \text{cancel out the denominators} \\[1 em]8 &= 9x-2x^2&& \text{simplify right side} \\[1 em]8 &= -2x^2+9x&& \text{move all terms to the left hand side } \\[1 em]8+2x^2-9x &= 0&& \text{simplify left side} \\[1 em]2x^2-9x+8 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}-9x+8 = 0 $ is a quadratic equation.
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