Every quadratic equation



has either

1. Two real solutions
2. One real solution
3. Two conjugate imaginary solutions

The quadratic equation:

  

has two real solutions when the " ± " part gives two different 
answers, one for the + and another for the -.  This will always 
be the case when the radicand, or "discriminant",

¹0 

If 

 > 0, the two solutions are real, and when

 < 0, the two solutions are conjugate imaginary
numbers.

However, if 

 = 0, there is but one real solution, because the
"±" part is "±0", and whether we add 0 or subtract 0, we still 
get the same number.