Question
$$\frac{4}{x}-3 = \frac{5}{2}x+3$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 6 }{ 5 }-\dfrac{ 2 \sqrt{ 19}}{ 5 } & x_2 = -\dfrac{ 6 }{ 5 }+\dfrac{ 2 \sqrt{ 19}}{ 5 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{4}{x}-3 &= \frac{5}{2}x+3&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2\cdot\frac{4}{x}-x\cdot2\cdot3 &= x\cdot2 \cdot \frac{5}{2}x+x\cdot2\cdot3&& \text{cancel out the denominators} \\[1 em]8-6x &= 5x^2+6x&& \text{move all terms to the left hand side } \\[1 em]8-6x-5x^2-6x &= 0&& \text{simplify left side} \\[1 em]-5x^2-12x+8 &= 0&& \\[1 em] \end{aligned} $$
$ -5x^{2}-12x+8 = 0 $ is a quadratic equation.
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