Find the number of roots of the equation `tan(x+pi/6)=2tanx ,forx in (0,3pi)dot`

Explanation

`tan(x+pi/6) = 2 tanx`

`=> (tanx+1/sqrt3)/(1-tanx*1/sqrt3) = 2 tanx`

`=>(sqrt3tanx+1)/(sqrt3 - tanx) = 2tanx`

`=>2tan^2x - sqrt3tanx+1 =0`

This is a quadratic equation.

So, its discriminant , `D = (sqrt3)^2 - 4(1)(2) = 3-8 = -5`

As `D lt 0`, no real roots exists for the given equation.