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.