Solving Trig equations: tan 2x + tan x - 2 = 0

Question

How to solve tan 2x + (tan (x)) - 2 = 0?

I already rewrote tan 2x in its double angle identity format, but I don't know where to go from there, or if I'm even doing it right.

Answer 1

Hint: Use that $$\tan(2x)=\frac{2\tan(x)}{1-\tan(x)^2}$$ Then we get$$-\tan(x)^3+2\tan(x)^2+3\tan(x)-2=0$$ Now substitute $$\tan(x)=t$$