The least positive non-integral solution of the equation $$\sin \pi \left( {{x^2} + x} \right) = \sin \pi {x^2}{\text{ is}}$$

Solution :
$$\eqalign{ & {\text{We have}},\,\,\sin \pi \left( {{x^2} + x} \right) = \sin \pi {x^2} \cr & \Rightarrow \pi \left( {{x^2} + x} \right) = n\pi + {\left( { - 1} \right)^n}\pi {x^2} \cr & \therefore {\text{Either }}{x^2} + x = 2m + {x^2} \cr & \Rightarrow x = 2m \in I \cr & {\text{or }}{x^2} + x = k - {x^2},\,{\text{where }}k\,{\text{is an odd integer}} \cr & \Rightarrow 2{x^2} + x - k = 0 \cr & \Rightarrow x = \frac{{ - 1 \pm \sqrt {1 + 8k} }}{4} \cr} $$
For least positive non-integral solution is $$x = \frac{1}{2},{\text{when }}\,k = 1$$