When a quadratic with integer roots appears on the board In a math class, the quadratic $x^2 + 10x + 20$ is written on the board. Each student goes to the board and increases or decreases either the linear or constant coefficient by $1$. After some time, $x^2 + 20x + 10$ is written on the board. Did a quadratic with integer roots necessarily appear on the board at some time during this process? Why or why not?

I'm not too sure where to start with this problem. I first wondered when a quadratic of the form $$x^2+ax+b=0$$
Is odd, but I'm not too sure even the answer to that question. It's obvious that the Discriminant $D=\sqrt{a^2-4b}$ has to be an integer. But do you know when that happens?
 A: A quadratic with integer roots has to appear on the board at some point. Consider the polynomials
$$(x+n)(x+1)=x^2+(n+1)x+n,\ n\in\Bbb Z$$
Clearly these polynomials have integer roots. Now plot a lattice of the polynomials $x^2+ax+b$:
   ^ b          x
   |           x
20 S---------+x
   |         x
   |        x|
   |       x |
   |      x  |
   |     x   |
   |    x    |
   |   x     |
   |  x      |
   | x       | a
10 +x--------F->
   x         20
  x|
 x |
x  10

The polynomial on the blackboard starts at S and finishes at F. Each student moves this polynomial one step in any orthogonal direction, so the journey from start to finish can be treated as a lattice walk.
Yet the polynomials of the form $(x+n)(x+1)$, marked by x's, form an infinite line which separates the $(a,b)$ plane into two regions, one containing S and one containing F. Any walk from S to F must therefore cross at least one of these x's, so at least one polynomial that appeared on the board must have had integer roots.
