how many triples of real numbers $(a,b,c)$ are there such that a,b,c are the roots of the equation

Question

How many triples of real numbers $(a,b,c)$ are there such that $a,b,c$ are the roots of the equation ??

$$x^3 +ax^2+bx+c=0$$

Answer 1

So you want to know how many $a, b, c$ satisfy $(x-a)(x-b)(x-c)=x^3+ax^2+bx+c$ which is equivalent to

$-abc=c,ab+bc+ac=b,a+b+c=-a$

Answer 2

Solving Jeff's equations: $$ \eqalign{a&= 0,b=0,c=0\cr a&=1, b=-2, c=0\cr a&=1, b=-1, c=1\cr a = b^2/2-1, c=-b^2 - b +2, &\text{ where } b^3 - 2 b+ 2 = 0\cr}$$