If $\sin^{2} x- \cos x=1/4$, then the value of $x$ between $0$ and $2\pi$ are :

Given

$\sin ^2x-\cos x=\dfrac 14$

$\implies 1-\cos ^2x-\cos x=\dfrac 14$

$\implies 4-4\cos ^2x-4\cos x=1$

$\implies 4\cos ^2x+4\cos x-3=0$

$\implies 4\cos ^2x+6\cos x-2\cos x-3=0$

$\implies 2\cos x(2\cos x+3)-1(2\cos x+3)=0$

$\implies (2\cos x-1)(2\cos x+3)=0$

$\implies \cos x=\dfrac 12$ and $\cos x\neq -\dfrac 32$

$\implies x=\dfrac \pi 3,\dfrac {5\pi}3$