Question
$$x^2+\frac{1}{x^2}+x^6+\frac{1}{x^6} = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^2+\frac{1}{x^2}+x^6+\frac{1}{x^6} &= 0&& \text{multiply ALL terms by } \color{blue}{ x^2x^6 }. \\[1 em]x^2x^6x^2+x^2x^6\cdot\frac{1}{x^2}+x^2x^6x^6+x^2x^6\cdot\frac{1}{x^6} &= x^2x^6\cdot0&& \text{cancel out the denominators} \\[1 em]x^{10}+x^2+x^{14}+\frac{1}{x^2} &= 0&& \text{multiply ALL terms by } \color{blue}{ x^2 }. \\[1 em]x^2\cdot1x^{10}+x^2\cdot1x^2+x^2\cdot1x^{14}+x^2\cdot\frac{1}{x^2} &= x^2\cdot0&& \text{cancel out the denominators} \\[1 em]x^{12}+x^4+x^{16}+1 &= 0&& \text{simplify left side} \\[1 em]x^{16}+x^{12}+x^4+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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