A solution is sought for an equation in the form

By direct verification,

Therefore, letting x = p - q we may also identify a = 3pq and b = p^3 - q^3.

The latter leads to a quadrtic equation in q^3:

solving which gives

so that

where the possibility of a negative q has been discarded as an unacceptable oddity. Quite similarly we can find that

Thus

Note that in the equation x^3 + ax = b both coefficients were assumed positive. Should a happen to be negative, the mathematicians of the 16^{th} century would solve x^3 = ax + b instead. The reason for this becomes clear from even a superficial inspection of the original *Ars Magna*.

### References

- F. J. Swetz, From Five Fingers to Infinity'', Open Court, 1996 (Third printing), p. 368