KöMaL 2016 C1368

Solve the equation ${[x]}^2+ {\{x\}}^2 + x^2 +2[x]\{x\}=4x-2x[x]-2x\{x\}-1,$ where $[x]$ denotes the greatest integer not greater than the number $x$, and $\{x\}$ denotes the difference between $x$ and $[x]$.