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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