pt$\Leftrightarrow \frac{1-cos2x }{2}+\frac{\sqrt{3}}{2}sin2x+1+cos2x=\frac{3+\sqrt{2}}{2}$
$\Leftrightarrow 1-cos2x+\sqrt{3}sin2x+2+2cos2x=3+\sqrt{2}$
$\Leftrightarrow \sqrt{3}sin2x+cos2x =\sqrt{2}$
$\Leftrightarrow sin(2x+\frac{\pi}{6})=\frac{\sqrt{2}}{2}\Leftrightarrow \left[ \begin{matrix} 2x+\frac{\pi}{6}=\frac{\pi}{4}+k2\pi\\ 2x+\frac{\pi}{6}=\frac{3\pi}{4}+k2\pi\end{matrix}{} \right.\Leftrightarrow \left[\begin{matrix} x=\frac{\pi}{24}+k\pi\\ x=\frac{7\pi}{24}+k\pi\end{matrix} {} \right. (k\in Z)$
Vậy pt có 2 họ nghiệm