Solving the sum-and-product problem: a geometric solution via the Pythagorean theorem

Abstract

This article addresses a foundational problem of algebra: finding two numbers x1 and x2 given their sum s and their product p (with s, p>0). This ancient problem, which lies at the historical origin of quadratic equations, was solved geometrically by Babylonian and Greek mathematicians. While Euclid's Elements provide a famous ‘completing the square’ construction, this paper presents an alternative geometric method. By interpreting x1 and x2 as the side lengths of a rectangle defined by its semiperimeter (s) and area (p), we construct the solution using only a simple geometric figure and repeated application of the Pythagorean theorem. This approach is highly accessible, even for students first encountering the Pythagorean theorem (e.g. in middle school). The method yields the solution as a nested radical, which is then shown to be algebraically equivalent to the standard quadratic formula, thus bridging concrete geometry with abstract algebra.