$$x^2+\frac{101}{100}x^4+\frac{1}{4}x^6-25 = 0$$
Answer
$$ \begin{matrix}x_1 = 1.84552 & x_2 = -1.84552 & x_3 = -0.92075+2.13793i \\[1 em] x_4 = -0.92075-2.13793i & x_5 = 0.92075+2.13793i & x_6 = 0.92075-2.13793i \end{matrix} $$
Explanation
$$ \begin{aligned} x^2+\frac{101}{100}x^4+\frac{1}{4}x^6-25 &= 0&& \text{multiply ALL terms by } \color{blue}{ 100 }. \\[1 em]100x^2+100 \cdot \frac{101}{100}x^4+100\frac{1}{4}x^6-100\cdot25 &= 100\cdot0&& \text{cancel out the denominators} \\[1 em]100x^2+101x^4+25x^6-2500 &= 0&& \text{simplify left side} \\[1 em]25x^6+101x^4+100x^2-2500 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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