How many quadratic polynomials with real coefficients $a{x}^2+bx+c=0$ are there such that the set of roots equals the set of coefficients, and all coefficients $a,b,c$ are non-zero? (For clarification: If the polynomial is $a{x}^2+bx+c$, $a\ne 0$, and the roots are $r$ and $s$, then the requirement is that $\{a,b,c\}=\{r,s\}$ and $a\ne 0,b\ne 0,c\ne 0$.)