Can we solve 3 equations:-
$$a+b+c = x$$
$$abc = y$$
$$ab+bc+ac = z$$
to get the value of $a,b$ and $c$
where $x,y,z$ are constants
As you can probably see I'm trying to find a solution to a cubic polynomial
I know other ways to solve the polynomial but I just want to see if we can find the solution using theses three equations
2026-03-27 11:45:12.1774611912
Solution to equations in 3 variable without forming a cubic polynomial
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This is neither more nor less than the cubic equation $$ t^3 - xt^2 + zt - y = 0 $$ whose roots for $t$ are exactly your $a,b,c$. And every cubic equation can be transformed into this form.
If there were a known method of solving your nonlinear system of equations that were appreciably easier than Cardano's formula (for an exact solution with radicals), that method would be widely publicized as the good method for solving cubics. Since this hasn't happened, it is safe to assume that no such method is known.
Even if you're given the equation system ab initio and want just to find numerical approximations to $a,b,c$, the easiest method is probably to form the cubic and employ root-finding methods such as Newton-Raphson.