Question

Number of solution of the equation ${\sin }^{4}x-{\cos }^{2}x\sin x+2{\sin }^{2}x+\sin x=0$ in $0\le x\le 3\pi$ is

Answer 1

${\sin }^{4}x-{\cos }^{2}x\sin x+2{\sin }^{2}x+\sin x=0$
${\sin }^{4}x-(1-{\sin }^{2}x)\sin x+2{\sin }^{2}x+\sin x=0$
${\sin }^{2}x({\sin }^{2}x+\sin x+2)=0$
$\Rightarrow \sin x=0$ ($\because D<0$ for ${\sin }^{2}x+\sin x+2)=0$)
For $0\le x\le 3\pi$
$\sin x=0\Rightarrow x=0,\pi ,2\pi ,3\pi$
Number of solutions are $4$.