Solve $x^2-2\sqrt{x}+1=0$ in $\mathbb{R}$ without using numerical methods?

Question

How do I solve this equation: $$x^2-2\sqrt{x}+1=0$$ in $\mathbb{R}$ without using numerical methods?

Note: I have used variable change by letting $y=\sqrt{x}$ I got this equation:$(y-1)(y^3+y^2+y-1)=0$, at this step I'm not able to complete the solution, and I do not want to use "Cardan" method.

Thank you for any help.

Answer 1

I don't see how you can do much with this except with Cardano or numerical methods. The real root is given by

$$ y = \frac{\sqrt[3]{17+3\sqrt{33}}-1}{3}-\frac{2}{3\sqrt[3]{17+3\sqrt{33}}} \doteq 0.54369 $$

Other than that's a lot of threes, I don't see any obvious trick. But maybe someone will surprise us.