Answer: 5.

Solution. If $a=0$ or $a=b=c$, then $a=b=c=0$. Otherwise: if $a=b \neq c$, then either $a=b=-1, c=0$, or $a=b=1, c=-2$; if $a \neq b=c$, then either $a=1, b=c=-0.5$; if $a=c \neq b$, then $a=c=c_{0}, b=1 / c_{0}$, where the number $c_{0}<0$ is the unique root of the equation $c^{3}+c=-1$.
Problem 4. How many triples of numbers $a, b, c$ exist, each of which is a root of the corresponding equation $a x^{2}+b x+c=0$?