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 =
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