Question
$$\frac{x}{x^2}+1 = \frac{9}{x}$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{x}{x^2}+1 &= \frac{9}{x}&& \text{multiply ALL terms by } \color{blue}{ x^2x }. \\[1 em]x^2x \cdot \frac{x}{x^2}+x^2x\cdot1 &= x^2x\cdot\frac{9}{x}&& \text{cancel out the denominators} \\[1 em]\frac{1}{x^2}+x^3 &= 9&& \text{multiply ALL terms by } \color{blue}{ x^2 }. \\[1 em]x^2\cdot\frac{1}{x^2}+x^2\cdot1x^3 &= x^2\cdot9&& \text{cancel out the denominators} \\[1 em]1+x^5 &= 9x^2&& \text{simplify left side} \\[1 em]x^5+1 &= 9x^2&& \text{move all terms to the left hand side } \\[1 em]x^5+1-9x^2 &= 0&& \text{simplify left side} \\[1 em]x^5-9x^2+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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Equations Solver