How do you know if a quadratic equation will have one, two, or no solutions?

Answer 1

It will always have two solutions: the question is are they real, complex, or a double root. Are you asking how does one know if the solutions are real?

Answer 2

One solution: When the lines intersect
No solutions: The same line
(There is no 2 solutions) It's infiently many solutions : when the lines are paralell