How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator? Find all solutions of the equation $\tan(x) = 2 + \tan(3x)$ where $0<x<2\pi$.
By replacing $\tan(3x)$ with $\dfrac{\tan(2x) + \tan(x)}{1-\tan(2x)\tan(x)}$ 
I've gotten to $\tan^3 (x) - 3 \tan^2 (x) + \tan(x) + 1 =0$.
I am not sure how to proceed from there without the use of a calculator.

$t=\tan x=1$ is easy to deal with
$$t^2-2t-1=0\implies -2t=1-t^2\implies\frac{2t}{1-t^2}=-1$$
$$\tan2x=-1=-\tan\frac\pi4=\tan\left(-\frac\pi4\right)$$
$$2x=n\pi-\frac\pi4$$ where $n$ is any integer 
