# Intuition for why the difference between 2×2−xx2−x+1\frac{2x^2-x}{x^2-x+1} and x−2×2−x+1\frac{x-2}{x^2-x+1} is a constant?

Why is the difference between these two functions a constant?

Since the denominators are equal and the numerators differ in degree I would never have thought the difference of these functions would be a constant.

Of course I can calculate it is true: the difference is $2$, but my intuition is still completely off here. So, who can provide some intuitive explanation of what is going on here? Perhaps using a graph of some kind that shows what’s special in this particular case?

Thanks!

BACKGROUND: The background of this question is that I tried to find this integral:

As a solution I found:

Whereas my calculusbook gave as the solution:

I thought I made a mistake but as it turned out, their difference was constant, so both are valid solutions.

Would you be surprised that the difference of $\dfrac{2x^2+x+1}{x^2}$ and $\dfrac{x+1}{x^2}$ is $2$?