Question
$$x^5-x^3 = \frac{3}{2}$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^5-x^3 &= \frac{3}{2}&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2x^5-2x^3 &= 2\cdot\frac{3}{2}&& \text{cancel out the denominators} \\[1 em]2x^5-2x^3 &= 3&& \text{move all terms to the left hand side } \\[1 em]2x^5-2x^3-3 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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