Is it possible to find these numbers without converting them into cubic equation?

Question

$abc = k$

$ab + bc + ac = l$

$a+b+c = m$

($k$, $l$ and $m$ are known. $a\geq b\geq c$, $x,y,z\in\mathbb{R}$)

Is it possible to solve this equation system for $a$, $b$ and $c$ by only manipulating the equations (squaring, cubing, adding, subtracting, cube rooting...)?

I'm sure that there's a way, here's a quote:

In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions. This was also shown by Ruffini in 1813 (Wells 1986, p. 59).

Answer 1

Certainly not.

If that was possible, we would have an alternative method of solving a cubic equation.

In a way, solving a cubic equation without solving a cubic equation :)

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And as it seems that the values are real (you can order them), you cannot do without the trigonometric approach, this is a "casus irreductibilis".

Answer 2

In general, I don't see how to avoid a cubic for complex solutions. For example, solve the system \begin{align*} abc & = 1,\\ ab+bc+ca & = 2,\\ a+b+c & = 3. \end{align*} The solutions are given by the roots of the cubic $$ c^3-3c^2+2c-1=0, $$ and quadratic equations for $a$ and $b$. Whatever we do otherwise, must involve these roots of the cubic.