Solve This Quadratics Question with Some Clever Geometry // [MATH CONTEST]
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We're looking at a beautiful little question from the 2002 ARML today: given two quadratics, x² + ax + b and x² + ax + b + 1, where both a and b are integers, how many values of b between 0 and 2002 (inclusive) are there that will result in integer roots for both quadratics.
This problem came up in class, and I'll be honest, it stumped me for a second. So I did what I always do when a question stumps me: I went concrete. Could I think of at least one example of a trinomial with the same quadratic and linear terms, but with constant terms one apart AND could I think of such an example with integer roots (meaning that the quadratics were factorable).
As I reflected on what the terms in a trinomial represent, I realized this was a variation an an optimization problem. The linear term represents the sum of the roots, and a very common quadratic optimization problem has you hold that sum constant as it represents something like a certain perimeter of fencing. Then your job is to maximize the area. Furthermore, that area is always maximized when you make the rectangle a square, and FURTHER furthermore, that square always area exactly one greater than when the rectangle has dimensions one larger and one smaller than the square form.
Armed with that realization and a few concrete examples, we can quickly come up with the forty-four possible values of b between 0 and 2002 that work (corresponding to the 44 perfect squares beginning at 1 and going through 1936, the last perfect square less than 2002).
Finally, we can use the quadratic formula to write about all solutions generally, and see that our 44 aren't just some 44 possibilities, but are the only 44 possibilities.
All and all, a really great question for students interested in math contests like the American Mathematics Competition, MATHCOUNTS, UIL Math or UIL Number Sense, and many many others. This could be a tough problem on the AMC8 or a medium problem on AMC10.
#mathcontest #mathcounts #artofproblemsolving #quadratics #algebra
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