Quadratics That Factor, Perhaps Too Well

INTRODUCTION AND REFORMULATION 1

The following problem has come up a number of times over the years, and I never remember how I did it the last time. So maybe the time has come to write it down-otherwise, if it were to happen again, I probably won't remember it at all. Of course, it's now different from how I remembered it.

Assume that b and c are integers. We all learned a long time ago that if x2 - bx + c factors over the integers, so does x2 + bx + c. All we do is use the opposites of the zeros. But what if we want to factor x2 + bx - c? Most of the time, this doesn't factor, but the example that always gets people started is x2 - 5x + 6 = (x - 3)(x - 2). As luck would have it, x2 - 5x - 6 also factors, since it equals (x - 6)(x + 1). The usual question is: Are there any other examples? Yes, x2 + 13x ± 30 works: (x + 10)(x + 3) or (x + 15)(x - 2). We shall examine how to generate an infinity of such examples, and in fact try to generate all of them.