Algebra is defined by Wikipedia as "a branch of mathematics concerning the study of structure, relation , and quantity. Together with geometry, analysis, combinatorics, and number theory," Many people disagree on the origins of algebra, some think it all started with the Greeks, others believe that it all goes further back to the Egyptians. In reality, it has not been long since algebra was invented, in fact, it can all be traced back to 820 AD.
In approximately 820 AD, a Persian mathematician by the name of Muhammad bin Musa Khwarizmi wrote a book called Al-Kitb al-mukhtaar f hsb al-abr wa'l-muqbala, Arabic for The Compendious Book on Calculation by Completion and Balancing. It would later be translated by Robert of Chester into a Latin book titled Liber algebrae et almucabala, hence the name "algebra".
Within the book, there are methods for solving for the positive roots of polynomial equations, and fundamental methods of manipulating the equation by moving terms in the equation. The book classifies all quadratic equations to six types, using equations described as "squares", "roots", and "numbers". With respect to modern notation these six types are:
squares equal roots (ax2 = bx)
squares equal number (ax2 = c)
roots equal number (bx = c)
squares and roots equal number (ax2 + bx = c)
squares and number equal roots (ax2 + c = bx)
roots and number equal squares (bx + c = ax2)
One must be confused at the sight of this, a skilled mathematician would immediately recognize that by allowing the use of the number 0 in a quadratic equation, all equations could be represented using ax2 + bx + c = 0. But keep in mind that this is 820 AD, and people back then had a limited knowledge of mathematical formulas. These six equations are used to represent quadratic equations where all numbers are positive and none zero.
The completion mentioned in the title of the book refers to moving a negative number or variable to the other side of the equation and changing it's sign. For example, 6x = 10x2 2x would become 4x = 10x2. Repeating this step would eventually result in an equation with no negative numbers.
The Balancing part of this book deals with removing positive numbers from both side of the equation in equal proportions, in other words, canceling them out. An example would be 2x + 6 = x + 6x2, which would be changed into x + 6 = 6x2. This ensures that there are only at most one of each square, root, and number.
Using these two methods any quadratic equations could be simplified into one of six possible styles, which became the birth of Algebra.