$$x^4+2+\frac{1}{x^4} = 0$$
Answer
$$ \begin{matrix}x_1 = 0.70711+0.70711i & x_2 = 0.70711-0.70711i & x_3 = -0.70711+0.70711i \\[1 em] x_4 = -0.70711-0.70711i & \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x^4+2+\frac{1}{x^4} &= 0&& \text{multiply ALL terms by } \color{blue}{ x^4 }. \\[1 em]x^4x^4+x^4\cdot2+x^4\cdot\frac{1}{x^4} &= x^4\cdot0&& \text{cancel out the denominators} \\[1 em]x^8+2x^4+1 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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