1. Zero A zero in the discriminant means that there is one solution, since it's ... plus/minus zero, which is just ...
2. Negative Negative means no real solutions, as the square root of a negative number is not real.
3. Positive If you have a positive number within the square root, it means that your solutions will be ... plus/minus whatever the square root is. One solution is the plus, and one solution is the minus, hence two solutions.
