Solve nonlinear simultaneous equations with 3 variables

Question

How to solve this system?

$$ X + Y + Z = A $$ $$ X Y Z = B $$ $$ X^2 + Y^2 + Z^2 = C $$

For example: $(A, B, C )= (1, 2, 3)$

Answer 1

Hint:

The l.h.s. of these equations are symmetric functions of X, Y, Z, of which the first and last are the elementary symmetric functions $s_1$ and $s_3$. Determine the value of the second elementary symmetric function $$s_2=XY+YZ+ZX.$$ Once this is solved, you know that $X,Y,Z,$ are the roots of the cubic equation $$t^3-s_1t^2+s_2t-s_3=0.$$ For your example, you should find the real root $2$ and the two nonreal cubic roots of unity.