6 1. Cubic Equations
Finally, the desired solution x of the cubic equation x3 + px + q = 0
is obtained from Cardano's formula
l
/
f y
/
/
/
/
A
y y
/ /
/ u
Figure 1.2. Depicted here is the geometric basis of the bi-
nomial equation (it +
v)3
= 3uv(u + v) +
(w3
+ f
3
), similar
to Cardano's presentation in his Ars Magna. The large cube
can be decomposed into two subcubes and three rectangular
parallelepipeds, all with side lengths w, v, and u + v.
If we apply this result to our problem x3 + x 6 = 0, we obtain
x =
W 3 + •», , i, - •«,
w
3V 3 V 3V 3 '
whose decimal value is approximately 1.634365.
1.3 In his Ars Magna Cardano also solved cubic equations involv-
ing quadratic terms.6 We have already seen, in the introduction, an
6 Ars Magna, Chapter XXIII.
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