Given a field $F$ with $char F \neq 2,3$ and an elliptic curve $E: y^2-(x^3+ax+b)$ I want to find the number of points of order $2$. (The given solutions say it is always exactly 3.) Let $P$ be a point of order $2$: If $P\oplus P=0$ then $P=-P$ which implies that the $y$ coordinate of $P$ is zero, is this true so far?
So if the $y$ coordinate is zero, we have to consider the number of solutions of $x^3+ax+b=0$. But this polynomial does not always have three distinct roots, therefore it is not true that there are always $3$ points of order $2$.