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 , \~P- 2

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 =