Question

Solve the following equation:

\[\tan^2 x + \left( 1 - \sqrt{3} \right) \tan x - \sqrt{3} = 0\]

Answer 1

\[\tan^2 x + (1 - \sqrt{3}) \tan x - \sqrt{3} = 0\]\[\Rightarrow \tan^2 x + \tan x - \sqrt{3} \tan x - \sqrt{3} = 0\]
\[ \Rightarrow \tan x (\tan x + 1) - \sqrt{3} (\tan x + 1) = 0\]
\[ \Rightarrow (\tan x - \sqrt{3}) (\tan x + 1) = 0\]

\[\Rightarrow (\tan x - \sqrt{3}) = 0\] or

\[(\tan x + 1) = 0\]
Now,
\[\tan x - \sqrt{3} = 0 \]
\[ \Rightarrow \tan x = \sqrt{3} \]
\[ \Rightarrow \tan x = \tan \frac{\pi}{3} \]
\[ \Rightarrow x = n\pi + \frac{\pi}{3}, n \in Z\]
And,
\[\tan x = - 1 \]
\[ \Rightarrow \tan x = \tan\left( - \frac{\pi}{4} \right) \]
\[ \Rightarrow x = m\pi - \frac{\pi}{4}, m \in Z\]