all these equations are in the form of ax^2 + bx + c = 0, where a, b, and c are some numbers. the discriminants of equations like this are equal to b^2 - 4ac. if the discriminant is negative, there are two imaginary solutions. if the discriminant is positive, there are two real solutions. if the discriminant is 0, there is one real solution.
x^2 + 4x + 5 = 0
b^2 - 4ac
4^2 - 4(1)(5)
16-20
-4, two imaginary solutions.
x^2 - 4x - 5 = 0
b^2 - 4ac
(-4)^2 - 4(1)(-5)
16 + 20
36, two real solutions.
4x^2 + 20x + 25 = 0
b^2 - 4ac
20^2 - 4(4)(25)
400 - 400
0, one real solution.
