How to solve parameters given 3 different types of information?

Question

How Do I solve this eqns?

$$x+y+z = A$$ $$xyz = B$$ $$x^2+y^2+z^2 = C$$

I have tried it in this way,,,

$$yz = B/x = P$$ $$y+z = A-x = Q$$ $$y(Q-y) = P$$ $$\implies y^2-Qy+p = 0$$

I can't figure out what to do next., Solving the this quadratic equation for y and z and putting the values in 3, doesnt seem to work. cause there will be sqrt in the eqn.

Answer 1

We have $$xy+yz+zx=\frac{1}{2}((x+y+z)^2-x^2-y^2-z^2)=\frac{1}{2}(A^2-C)=C'$$ so by Vieta's formulas $x,y$ and $z$ are solutions of the polynomial equation: $$t^3-At^2+C't-B=0$$

Answer 2

Hint: $xy + yz + zx = (x+y+z)^2 - (x^2 + y^2 + z^2) = \frac{A^2 - C}{2}$

Consider the equation $M^3 - AM^2 + ( \frac{A^2-C} {2} )M - B=0$. It has roots $x, y$ and $ z$.