Question
$$-x^6+x^5+7x^2-\frac{9}{x^4} = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} -x^6+x^5+7x^2-\frac{9}{x^4} &= 0&& \text{multiply ALL terms by } \color{blue}{ x^4 }. \\[1 em]-x^4x^6+x^4x^5+x^4\cdot7x^2-x^4\cdot\frac{9}{x^4} &= x^4\cdot0&& \text{cancel out the denominators} \\[1 em]-x^{10}+x^9+7x^6-9 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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Polynomial Equations Solver