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[Karl Vierstein]

$sin2x+4tanx=\frac{9\sqrt{3}}{2}$

[duongtoi]

Đặt $\tan x=t$. Suy ra, $\sin 2x=\frac{2t}{1+t^2}$.

Vậy $PT\Leftrightarrow\frac{2t}{1+t^2}+4t=\frac{9\sqrt3}{2}$

$\Leftrightarrow 8t^3-9t^2\sqrt3+12t-9\sqrt3=0$

$\Leftrightarrow (t-\sqrt3)(8t^2-t\sqrt3+9)=0$

$\Leftrightarrow t=\sqrt3\Leftrightarrow x=\frac{\pi}{3}+k\pi.$

Vậy PT có nghiệm là $x=\frac{\pi}{3}+k\pi.$