Question
$$\frac{x}{x+2}+\frac{1}{x-2} = \frac{3}{x}$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{x}{x+2}+\frac{1}{x-2} &= \frac{3}{x}&& \text{multiply ALL terms by } \color{blue}{ (x+2)(x-2)x }. \\[1 em](x+2)(x-2)x \cdot \frac{x}{x+2}+(x+2)(x-2)x\cdot\frac{1}{x-2} &= (x+2)(x-2)x\cdot\frac{3}{x}&& \text{cancel out the denominators} \\[1 em]x^3-2x^2+x^2+2x &= 3x^2-12&& \text{simplify left side} \\[1 em]x^3-x^2+2x &= 3x^2-12&& \text{move all terms to the left hand side } \\[1 em]x^3-x^2+2x-3x^2+12 &= 0&& \text{simplify left side} \\[1 em]x^3-4x^2+2x+12 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver