Motivation Behind Tschirnhaus Transformation and Cardano's Formula I have a general question about the cubic equation:


  Let $a,b,c \in \mathbb{C}$ with $a \neq 0$. The general cubic equation is $t^3+at^{2}+bt+c = 0$. To get Cardano's Formula, we first transform the equation so that $a= 0$ (i.e. $t^3+bt+c = 0$). Then we reduce this to a quadratic that we can solve. What is the motivation behind the transformations that reduce $t^3+at^2+bt+c = 0$ to $t^3+bt+c = 0$?


In particular, if we let $y = t+ \frac{a}{3}$, then $t = y-\frac{a}{3}$ which cancels the $at^2$ term. This is called a Tschirnhaus transformation. We are left with an equation of the form $$y^3+py+q = 0$$

But how do we know what transformations to use to get Cardano's Formula, where $$p = \frac{a^2-2a^3+3b}{3}\quad\mathrm{and}\quad q = \frac{2a^3-9ab+27c}{27}\quad ?$$ Finally if we let $y = \sqrt[3]{u}+ \sqrt[3]{v}$ we eventually get a quadratic which leads to Cardano's Formula.
I don't think it's really all that mysterious. The motivation behind using the Tschirnhaus Transformation seems clear; it's better to have fewer terms and fewer coefficients when solving an equation. It next probably occurred to Cardano to try $y = u + v$ and see what happens. Rewrite the equation as
$$u^3 + v^3 + 3uv(u + v) + p(u + v) + q = 0$$
You have a little leeway in choosing $u$ and $v$, since you only need them to sum to a root of the equation. So you might want to impose a second condition on $u$ and $v$ to simplify things (and two equations in the two variables $u$ and $v$ should specify their values). To reduce the above to a quadratic equation, you can stipulate that $uv = -{p \over 3}$. Namely you have
$$u^3 + v^3 + q = 0$$
which is equivalent to
$$u^3 -{p^3 \over 27}{1 \over u^3} + q = 0$$
Or just
$$u^6 + qu^3 - {p^3 \over 27} = 0$$
This is a quadratic equation in $u^3$, which you can solve. $u$ can be any of the three cube roots of this, and $v$ is determined by the equation $3uv = -p$. Thus you have three possible values of $u + v$ which will be the three roots of your equation.

In Cardano's time they didn't really know much about complex numbers (i.e. that every number has $n$ complex $n$th roots) but he was smart enough to basically understand what was going on.
That bugs me a little too, but I think the great advantage is this one: by making $t=u+v$, $y=t+k$ or whatever, you are giving your problem more degrees of freedom. Since there is more than one tuple $(u,v)$ that satisfies this relation, you will be able to impose another condition on the new coefficients in order to make the problem similar to something you can solve. By annulating one of them, for example.

It is also interesting to think the other way around. At first you don't know how to solve $ax^2+bx+c=0$ (I). However, you can always solve the simpler equation $x^2-p=0$ (II). If you can prove that by substituting $x$ for $[mx+n]$ in (II) you can always arrive at (I) for appropriate $m$, $n$ and $p$, you'll also be able to solve (I). Problem is: in this case we have 3 equations - one for comparing each coefficient - and three variables. As the degree gets bigger*, we keep having three variables but the number of equations increases, yielding the possibility of no solutions to the system.

*$a_0+a_1 x+\dots+a_k x^k=0 = (mx+n)^k-p=0$