2 1. Cubic Equations the quantity (|) is added to both sides, and the q is moved to the other side of the equation, yielding *+-+(§)'=(§)'-* Now the left-hand side of the equation can be represented as a square: And by taking square roots, one obtains the general solution formula P , \~P2 x1,2 = - - ] - - q . Note the following important property of quadratic equations: if one forms the negative sum of the two solutions and their product, one obtains the coefficients of the original equation, namely, x\ -+- x2 = p and Xix2 = q. Such completions of the square, in the form of geometric manip- ulations, were known already to the Babylonians around 1700 BCE. Quadratic equations were treated systematically in the works of the Baghdad scholar al-Khwarizmi (ca. 780-850), which were later trans- lated into Latin, inspiring mathematical progress in Europe for cen- turies. His name is the origin of the word algorithm. Moreover, the word algebra is derived from the title of one of his works, al-Jabr. From the modern point of view, al-Khwarizmi's method of han- dling quadratic equations is quite cumbersome. All statements and proofs are expressed in words, without algebraic symbols, which had not yet been invented. Furthermore, all the argumentation is of a geometric nature. And finally, since negative numbers had not been discovered—and no wonder, given the geometric context—al- Khwarizmi had to distinguish various types of equations, which today we would notate as x2 = px, x2 = q, x2 + q = px, x2 + px = g, and x2 = px + q, and since we have no difficulty accepting coefficients that are less than or equal zero, we can easily reduce all these to a single type. Figure 1.1 gives an impression of the method of argumentation used by al-Khwarizmi. From the figure one can see that the desired side length x of the inner square can be calculated from area q =
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