$$\frac{2}{x-2}+3x = \frac{4}{5}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 17 }{ 15 }-\dfrac{\sqrt{ 19 }}{ 15 } & x_2 = \dfrac{ 17 }{ 15 }+\dfrac{\sqrt{ 19 }}{ 15 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{2}{x-2}+3x &= \frac{4}{5}&& \text{multiply ALL terms by } \color{blue}{ (x-2)\cdot5 }. \\[1 em](x-2)\cdot5\cdot\frac{2}{x-2}+(x-2)\cdot5\cdot3x &= (x-2)\cdot5\cdot\frac{4}{5}&& \text{cancel out the denominators} \\[1 em]10+15x^2-30x &= 4x-8&& \text{simplify left side} \\[1 em]15x^2-30x+10 &= 4x-8&& \text{move all terms to the left hand side } \\[1 em]15x^2-30x+10-4x+8 &= 0&& \text{simplify left side} \\[1 em]15x^2-34x+18 &= 0&& \\[1 em] \end{aligned} $$
$ 15x^{2}-34x+18 = 0 $ is a quadratic equation.
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