Number of real values of x, satisfying the equation ${\left[x\right]}^2-5\left[x\right]+6-\sin x=0$ ( $\left[·\right]$ denotes G.I.F )

$\text{ roots, }\left[x\right]=\frac{5\pm \sqrt{25+4\sin x-24}}{2}=\frac{5\pm \sqrt{1+4\sin x}}{2}$

$-1\le \sin x\le 1\Rightarrow -4\le 4\sin x\le 4$

$\Rightarrow -3\le 1+4\sin x\le 5$

$\Rightarrow 0\le 1+4\sin x\le 5$

$\Rightarrow \left[x\right]\text{ is an integer }\iff \sin x=0\Rightarrow x=\pi$

$\Rightarrow x=3$