Solve algebraic equation with floor functions I am looking for a way to find all the solutions to this equation without guess and check:
$$\frac{-x^2+45x}{2x+1}-\left\lfloor \frac{-x^2 +45x}{2x+1} \right\rfloor + x - \lfloor x \rfloor = 0$$
I graphed it in desmos and found that the most obvious solutions are $x = -4, -1, 0, 3$. But it turns out that $x = 6$ is also a solution. I can't figure out how to get these solutions algebraically. If they can't be solved directly, does anyone know a numerical algorithm that can be used to at least estimate the solutions?
Also, are algebraic equations involving floor functions generally solvable with algebra, or is it only special cases?

$$\frac{-x^2+45x}{2x+1}-\left\lfloor \frac{-x^2 +45x}{2x+1} \right\rfloor + x - \lfloor x \rfloor = 0$$
The solution comes from $$\frac{-x^2 +45x}{2x+1}=-x+n$$ Where n is an integer.
Cross multiplication  gives us
$$-x^2 +45x=(2x+1)(-x+n)$$ or $$ x^2+(46-2n)x+n=0$$
This equation has real solution for  $$(46-2n)^2-4n\ge 0$$
