Two consecutive positive integers have the property that the sum of their squares is 91 more than their product. What are these integers?

Explanation

Let the two consecutive positive integers be x and x + 1. According to the problem, the sum of their squares minus their product equals 91: x² + (x + 1)² - x(x + 1) = 91. Simplifying, we get x² + x² + 2x + 1 - x² - x = 91, which reduces to x² + x + 1 = 91. Rearranged, this becomes x² + x - 90 = 0. Factoring yields (x + 10)(x - 9) = 0, so x = -10 or x = 9. Since the integers are positive, x = 9. Therefore, the integers are 9 and 10.