Solution: Since the distance between the roots is $\mathrm{D}$, the equations $\mathrm{f}(\mathrm{x})=0$ and $\mathrm{f}(\mathrm{x}-\mathrm{D})=0$ have a common root. If we draw the graphs of these functions, they will intersect the x-axis at the same point (this root). The vertical line passing through this point is the axis of symmetry of this drawing, and thus also for the graph of the sum. Therefore, this root is unique. There is also a purely computational solution to this problem.

Answer: one.
4. Let $f(x)=x^{2}+b x+c$. It is known that the discriminant of the quadratic equation $f(x)=0$ is 2020. How many roots does the equation $f(x-2020)+f(x)=0$ have?