3cos^2x+4sinxcosx+5sin^2x=2

[latex]3cos ^{2} x+4sinxcosx+5sin ^{2} x=2[/latex] [latex]3cos ^{2} x+4sinxcosx+5sin ^{2} x-2(sin ^{2} x+cos ^{2} x)=0[/latex] [latex]3cos ^{2} x+4sinxcosx+5sin ^{2} x-2sin ^{2} x-2cos ^{2} x=0[/latex] [latex]cos ^{2} x+4sinxcosx+3sin ^{2} x=0 [/latex] |:[latex]cos ^{2} x,[/latex]  [latex](cos ^{2} x \neq 0)[/latex] [latex]1+4 \frac{sinx}{cosx} +3 \frac{sin^{2}x }{cos ^{2}x } =0[/latex] [latex]1+4tgx+3tg ^{2} x=0[/latex] [latex]tgx=y[/latex] [latex]3y ^{2} +4y+1=0[/latex] D = 4² - 4· 3·1 = 16 - 12 = 4 [latex] \sqrt{D} =4[/latex] [latex]y _{1} = \frac{-4+2}{6} = \frac{-2}{6} =- \frac{1}{3} [/latex] [latex]y _{2} = \frac{-4-2}{6} = \frac{-6}{6} =-1[/latex] [latex] \left \{ {{tgx=- \frac{1}{3} } \atop {tgx=-1}} \right. [/latex] [latex]x=-arctg \frac{1}{3} + \pi n,[/latex] n∈Z [latex]x=- \frac{ \pi }{4} + \pi m,[/latex] m∈Z