Giải pt sau đây: [tex]sin2x-2\sqrt{2}(sinx+cosx)-5=0[/tex]
@iceghost
Đặt $t = \sin x + \cos x = \sqrt{2} \sin (x + \dfrac{\pi}4)$ ($|t| \leqslant \sqrt{2}$)
Suy ra $t^2 = 1 + \sin 2x$
pt $\iff t^2 - 1 - 2\sqrt{2} t - 5 = 0$
$\iff t^2 - 2\sqrt{2}t + 2 = 8$
$\iff (t - \sqrt{2})^2 = 8$
$\iff t - \sqrt{2} = 2\sqrt{2}$ hoặc $t - \sqrt{2} = -2\sqrt{2}$
$\iff t = 3\sqrt{2}$ (L) hoặc $t = -\sqrt{2}$
$\iff \sin(x + \dfrac{\pi}4) = -1$
$\iff x + \dfrac{\pi}4 = -\dfrac{\pi}2 + k2\pi$
$\iff x = -\dfrac{3\pi}{4} + k2\pi \ (k \in \mathbb{Z})$