$$\frac{1}{x+5}-\frac{x^2}{x^2-65} = 0$$
Answer
$$ \begin{matrix}x_1 = -5.88 & x_2 = 0.94+3.18917i & x_3 = 0.94-3.18917i \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x+5}-\frac{x^2}{x^2-65} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x+5)(x^2-65) }. \\[1 em](x+5)(x^2-65)\cdot\frac{1}{x+5}-(x+5)(x^2-65)\frac{x^2}{x^2-65} &= (x+5)(x^2-65)\cdot0&& \text{cancel out the denominators} \\[1 em]x^2-65-(x^3+5x^2) &= 0&& \text{simplify left side} \\[1 em]x^2-65-x^3-5x^2 &= 0&& \\[1 em]-x^3-4x^2-65 &= 0&& \\[1 em] \end{aligned} $$
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