I was given a question in a test
Let $\alpha$ be a real root of the equation, $x^5 – x^3 + x – 2 = 0$. Then the value of $[\alpha^6]$ is (where [*] denotes the greatest integer function)
the solution given is
(Side note): the second line uses sum of geometric progression with first term x
But I can't understand, why would anyone, after reading the question, think of $\sqrt{2}$ for checking the value of function. Is there any way of guessing what values of $x$ can be useful for getting a useful inequality.
Or is there any other better way to find the value of $\lfloor x \rfloor$?