$\sqrt3\sin x+\cos x=\sqrt2$
$\to 2\sin\left(x+\dfrac{\pi}{6}\right)=\sqrt2$
$\to \sin\left(x+\dfrac{\pi}{6}\right)=\dfrac{\sqrt2}{2}$
$\to \left[ \begin{array}{l}x+\dfrac{\pi}{6}=\dfrac{\pi}{4}+k2\pi\\x+\dfrac{\pi}{6}=\dfrac{3\pi}{4}+k2\pi \end{array} \right.$
$\to \left[ \begin{array}{l}x=\dfrac{\pi}{12}+k2\pi \\x=\dfrac{7\pi}{12}+k2\pi\end{array} \right.$
