$$\frac{1}{2}x^4-x = 0$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = 1.25992 & x_3 = -0.62996+1.09112i \\[1 em] x_4 = -0.62996-1.09112i & \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{2}x^4-x &= 0&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{1}{2}x^4-2x &= 2\cdot0&& \text{cancel out the denominators} \\[1 em]x^4-2x &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ x^{4}-2x = 0 } $, first we need to factor our $ x $.
$$ x^{4}-2x = x \left( x^{3}-2 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The remaining roots can be found by solving equation $ x^{3}-2 = 0$.
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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