X
Prefaces to the German Editions
If we apply this formula to our example, we obtain the two solu-
tions
xi = 3 + 2\/2 and x2 = 3 - 2\/2.
If you are interested in a numerical solution, you can pull out
your handy pocket calculator (or perhaps you know how to com-
pute square roots by hand) and obtain the decimal representations
x\ 5.828427... and x2 = 0.171572 You could also use your cal-
culator to verify that these values are in fact solutions to the original
equation. A skeptic who wished to verify that the solutions derived
from the formula are the exact solutions would have to substitute
the expressions containing the square roots into the equation and
demonstrate that the quadratic polynomial x2 6x + 1 = 0 actually
vanishes—that is, assumes the value zero—at the values x = x\ and
x = x2-
The Solution of Equations of Higher Degree. It has long been
known how to solve cubic equations such as
x3
-
3x2
- 3x - 1 = 0
by means of a formula similar to the quadratic formula. Indeed,
such formulas were first published in 1545 by Cardano (1501-1576)
in his book Ars Magna. However, they are quite complicated, and
have little use for numerical calculation. In an age of practically
unlimited computing power, we can do without such explicit formulas
in practical applications, since it suffices completely to determine the
solutions by means of numeric algorithms. Indeed, for every such
equation in a single variable there exist approximation methods that
iteratively, that is, step by step, compute the desired solution more
and more precisely. Such a procedure is run until the solution has
reached an accuracy suitable for the given application.
However such iterative numeric procedures are unsuitable when
not only the numerical value of a solution is sought, such as x\
3.847322... in the previous example, but the "exact" value
x1 = 1 + ^ 2 + ^ 4 .
It is not only that such an algebraic representation possesses a certain
aesthetic quality, but in addition, a numeric solution is insufficient if
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