
Solution by Radicals 27
where α,β are specific choices of the cube roots.
However, not all of these expressions are zeros. Equation (1.5) implies (1.7), but
(1.7) implies (1.5) only when we make the correct choices of cube roots. If we choose
α,β so that 3αβ + p = 0, then the solutions are
α + β ωα + ω
2
β ω
2
α + ωβ
Another peculiarity emerges when we try to solve equations whose solutions we
already know. For example,
y
3
+ 3y −36 = 0
has the solution y = 3. Here p = 3,q = −36, and Cardano’s formula gives
y =
3
q
18 +
√
325 +
3
q
18 −
√
325
which seems a far cry from 3. However, further algebra converts it to 3: see Exercise
1.4.
As Cardano observed in his book, it gets ...