How did Euclid solve quadratic problems through the application of areas?

Euclid solved quadratic problems through a procedure known as the application of areas, which is presented in Book VI of his work, the Elements.12

Rather than using modern symbolic algebra or equations, which were alien to the Greek mathematical spirit, Euclid utilized a geometric approach. This method involved constructing figures—specifically rectangles and parallelograms—to represent algebraic identities and solve problems of the second degree. By applying the theory of proportions, Euclid could manipulate these geometric magnitudes to find solutions that are equivalent to solving quadratic equations in modern terms.312

It is important to note that for the ancient Greeks, this was strictly a geometric process. They maintained a fundamental requirement for homogeneity, meaning they only compared like magnitudes (such as line to line or area to area) and did not possess the concept of an algebraic equation or symbolic manipulation. Consequently, while these constructions are now recognized as solving quadratic problems, they were conceived by Euclid as rigorous geometric proofs regarding the properties of areas and proportions.312