Problem: Otto Neugebauer believes that the Babylonians were quite capable of reducing the general cubic equation to the "normal form" $n^3 + n^2 = c$, although there is as yet no evidence that they actually did do this. Show how such a reduction might be made.
Eves Remark: Take the cubic in $x$ with unit leading coefficient and subject it to a linear transformation of the type $x = y+m$. Determine $m$ so that the resulting cubic in $y$ lacks the linear term.
External remark: One can bring every cubic equation of the form $x^3+ax^2+bx+c = 0$ into the form $n^3 + n^2 = p$. The transformation to $u^3+qu^2 = r$ comes from the substitution $x=u+s$ and $s$ can be determined from a quadratic equation [my comment: how is this helpful? What quadratic equation exactly?]. If $u^3 + qu^2 = r$ is divided by $q^3$, then $(\frac{u}{q})^3 + (\frac{u}{q})^2 = \frac{r}{q^3}$.
My remark: Concerning Eves' remark, I cannot determine a satisfactory value for $m$. I end up with a very messy expression, just like I do when I try to use the advice in the "external remark." If I make the substitution $x=u+s$ and expand the general cubic, I get the following: $s^3+s^2(3u+a)+s(3u^2+2au+b)+u^3+au^2+bu+c=0$. This does not seem to be of much help; the remark about being able to solve for $s$ using a quadratic equation would be helpful if this were further explicated, but I cannot find a satisfactory value for $s$. Any ideas about how to solve the originally stated problem?
