Solutions of $x^2-6x+\lfloor x \rfloor+7=0$ 
What are the roots of $x^2-6x+\lfloor x \rfloor+7=0$, where $\lfloor x \rfloor$ is the greatest integer function?  

Is there some way to solve the equation without graphing?

$x-1 < [x] \le x$, so $x^2-5x+6<0$ while $x^2-5x+7\ge 0$, which furthur reduces to $2<x<3$. We know $[x]=2$, so $x^2-6x+9=0$ and $x=3$. But this contradicts $x<3$ and hence no solution.
