http://www.targetiit.com/iit-jee-forum/posts/greatest-integer-function-18354.html
see my answer there
$\textbf{Calculate value of $\mathbf{x}$ in $\mathbf{2\cdot \left\{x\right\}=3[x]-[x^2]}$ }\\\\ \textbf{Where $\mathbf{\left\{.\right\}$= Fractional Part of $\mathbf{x}$}}\\\\ \textbf{and $\mathbf{[.]=}$ Integer Part}$
Hint:
RHS is integer...
LHS is either 0 or 1...
So x=n or x=n+1/2
Now proceed....
http://www.targetiit.com/iit-jee-forum/posts/greatest-integer-function-18354.html
see my answer there