$(\sin x+\cos x)^2-\sqrt3\cos 2x=3$
$\Rightarrow {\sin }^2x+2\sin x\cos x+{\cos}^2x-\sqrt3\cos2x=3$
$\Rightarrow 1+\sin2x-\sqrt3\cos2x=3$
$\Rightarrow \dfrac{1}{2}\sin 2x-\dfrac{\sqrt3}{2}\cos 2x=\dfrac{2}{2}=1$
$\Rightarrow \sin(2x-\dfrac{\pi}{3})=1$
$\Rightarrow2x-\dfrac{\pi}{3}=\dfrac{\pi}{2}+k2\pi(k\in\mathbb Z)$
$\Rightarrow x=\dfrac{5\pi}{12}+k\pi(k\in\mathbb Z)$
Nghiệm dương nhỏ nhất khi $k=0\Rightarrow a=\dfrac{5\pi}{12}$
Nghiệm âm lớn nhất khi $k=-1\Rightarrow b=\dfrac{-7\pi}{12}$
$\Rightarrow a+b=\dfrac{-\pi}{6}$
