Floors, Fractions, and a Quadratic Twist

Consider the equation $$3\lfloor x\rfloor^2 = x\lfloor x\rfloor + 21\{x\}^2,$$ where $\lfloor x\rfloor$ and $\{x\}$ are the greatest integer part of $x$ and the fractional part of $x$, respectively. The sum of all solutions to the equation can be written in the form $p/q$, where $p$ and $q$ are relatively prime positive integers. What is the value of $p+q$?