3 ctg^2x = 2tg^x - 5

3ctg²x-2 tg²x+5=0 ,  ctgx=1/tgx  [latex]\frac {3}{tg^{2}x} -2tg^{2}x+5=0 , \\ tgx=t \\ \frac {3}{t^{2}} -2t^{2} +5=0 , \\ 3-2t^{4} +5t^{2} =0 , t\neq 0 \\t^{2}=y \\2y^{2} -5y-3=0 \\D=25+24=49 , \\y_1 =-\frac {-1}{2} , y_2=3 \\t^{2}\neq-\frac{1}{2} , t^{2}=3 \\ t_1=-\sqrt{3} , t_2=\sqrt{3} \\tgx=-\sqrt{3} , x=arctg(-\sqrt{3})+\pi n , x=-\frac{\pi}{3}+\pi n \\tgx-\sqrt{3} , x=arctg\sqrt{3}+\pi k, x=\frac{\pi}{3}+\pi k[/latex]  n∈Z , k∈Z.