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

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`tan (x+pi/6)=2tan x`
`rArr (tan x +"tan" pi/6)/(1- tan x "tan" pi/6)=2 tan x`
Let `tan x=y`.
`rArr (y+1/sqrt(3))/(1- y/sqrt(3))=2y`
`rArr (sqrt(3) y+1)/(sqrt(3)-y)=2y`
`rArr sqrt(3) y+1=2sqrt(3) y-2y^(2)`
`rArr 2y^(2)-sqrt(3)y+1=0`, which has imaginary roots.
Hence, equation has no solutions.
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