Geometric techniques were widely used across ancient civilizations, including Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation, demonstrating the Egyptians' advanced understanding of these mathematical concepts. Engineers from these cultures understood how the area of a square scales with the length of its side, such as knowing that tripling the side length of a square results in nine times the area. They also calculated areas of more complex shapes like rectangles and T-shapes.

By around 400 BC, Babylonian mathematicians and, by 200 BC, Chinese mathematicians employed geometric methods of dissection to solve quadratic equations with positive roots. These methods laid the groundwork for more abstract algebraic techniques developed later.

The quadratic equation, as it is recognized today, was significantly advanced by the Persian mathematician Muhammed ibn Musa al-Khwarizmi (circa 815-850). Under the patronage of the Caliph, al-Khwarizmi compiled all available knowledge on algebra and authored the first comprehensive text on the subject. His work, which included methods for solving quadratic equations, marked a pivotal moment in the history of mathematics. The title of his book, which included the term "algebra," signified the birth of this new branch of mathematics. Al-Khwarizmi's contributions provided systematic approaches to solving first-degree equations and simple quadratics, which required only square roots for their solutions, before delving into more complex quadratic equations.

In summary, the history of quadratic equations spans several millennia and multiple civilizations, with significant contributions from Babylonian, Egyptian, Chinese, and Islamic mathematicians. Their combined efforts have shaped the understanding and methods used to solve quadratic equations, forming a cornerstone of modern algebra.

*The Evolution of Quadratic Equations: A Historical Perspective*