The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following:[37], and, to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:[37], Arithmetica is a collection of some 150 solved problems with specific numbers and there is no postulational development nor is a general method explicitly explained, although generality of method may have been intended and there is no attempt to find all of the solutions to the equations. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. New York: Wiley, 1968.

On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems. Euclid (Greek: Εὐκλείδης) was a Hellenistic Egyptian mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). [8] One of the most famous tablets is the Plimpton 322 tablet, created around 1900 - 1600 BCE, which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek mathematics. By 750 Arab armies had established control of a region extending from southern Spain through North Africa, Asia Minor, and into India.

With Viète's new notation, it became easier to think of solving an algebraic equation as finding the values of x for which a definite function of the variable x would equal zero. where p and q are positive.

The Byzantine Empire had, however, provided a refuge for Greek-speaking scholars. "Al-Khwarizmi, Abu Jafar Muhammad Ibn Musa." His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. [20] By the time of Plato, Greek mathematics had undergone a drastic change. It has also been suggested that the idea came via China. [44] In indeterminate analysis Brahmagupta gives the Pythagorean triads , , , but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with. An example of geometric algebra would be solving the linear equation ax = bc. [8] They were familiar with many simple forms of factoring,[8] three-term quadratic equations with positive roots,[10] and many cubic equations[11] although it is not known if they were able to reduce the general cubic equation. As shown in this graph, to solve the third-degree equation where Omar Khayyám constructed the parabola the circle with diameter having its center on the positive x-axis and intersecting the origin, and a vertical line through the point above the x-axis where the circle and parabola intersect. During the Dark Ages, European mathematics was at its nadir with mathematical research consisting mainly of commentaries on ancient treatises; and most of this research was centered in the Byzantine Empire. astronomy, math…, Development Fund for Black Students in Science and Technology, Development Doctrine and Modernization Theory, development and growth: school age and adolescence, development and growth: birth and infancy, Developing Design Principles to Scaffold ePBL: A Case Study of eSTEP, Developing Creative Learning Environments in Problem-based Learning, Developers Diversified Realty Corporation, DeVeaux, Nathaniel (Nathaniel De Veaux, Nathaniel Deveaux), Development of Commercial Banking 1950–1990, Development of Higher-Dimensional Algebraic Concepts, Development of Physical Chemistry during the Nineteenth Century, Development of Prenatal Diagnostic and Surgical Techniques, Development of Processes Underlying Learning, Development of Seagoing Vessels in the Ancient World, Development of Shanghai's Advanced Manufacturing Industry, Development of Shanghai's Modern Service Industry, Development of the Fundamental Notions of Functional Analysis, Development Superintendency of the Northeast (SUDENE). "[52] Al-Khwarizmi's work established algebra as a mathematical discipline that is independent of geometry and arithmetic.

After further development among Hellenistic and Indian mathematicians, it was eventually the work of Islamic mathematicians that established algebra as an independent discipline in its own right. And it is here that Algebra was further developed. For instance, proposition 6 of Book II gives the solution to the quadratic equation , and proposition 11 of Book II gives a solution to .