How to solve $x^3-3x+3=0$?

Question

I have the following cubic equation. Is it possible to solve it analytically in very "easy" way to a student only has a pre-calculus level:

The equation is:

$$x^3-3x+3=0$$

The real root I want to show is $x=-2.1038$

Update: Cardono's method is most suitable for my case so far, but I want to check if something is easier. Thank you all.

Aside note: I believe this cubic problem is unsuitable for students not exposed to numerical methods.

Answer 1

Using rational root theorem, it is easy to check that the equation has no rational root.

A good pre-calculus method is the bisection method. The given expression is $>0$ for $x=-2$ and $<0$ for $x=-3.$ So, we know there is a root between $-2$ and $-3$. Now simply apply the bisection method.