Three Kinds of Answers

The discriminant sits under the square root in the formula, and a square root behaves in three ways depending on the sign of what is under it. So, when a, b and c are real, the sign tells you what the answers look like before you solve anything.

Positive: the root is a real number and the ± gives two different real answers. x^2 - 6x + 5 = 0 has discriminant 16, and its solutions are 1 and 5. Zero: the root is 0, so adding it and subtracting it give the same number. x^2 + 2x + 1 = 0 has discriminant 4 - 4, and its only solution, -1, is called a repeated solution.

Negative: the root of a negative number needs i, so both answers are complex. For x^2 + 2x + 5 = 0 the discriminant is 4 - 20, which is -16, and the formula gives (-2 ± √(-16))/2: x = -1 + 2i and x = -1 - 2i. The ± makes them conjugates, with the same real part and opposite imaginary parts. When a, b and c are real, complex solutions always come in such a pair.