Hello from MrBillDoesMath!
Answer:
0, 1, or 2
Discussion:
Examples:
1) x^2 + 9 = 0 has 0 solutions over the real numbers
2) x^2 + 9 = 0 has 2 solutions over the complex number ( 3i, -3i)
3) the quadratic (x-3)^2 = 0 has a single solution, x = 3, of multiplicity two. (which some regard as two solutions).
The bottom line is the number of roots is given by the quadratic solution
( -b +\- sqrt( b^2 - 4ac) ) /2a
In particular if the discriminant (b^2 -4ac) = 0, the quadratic has one solution. If the discriminant is >0 there are two (real) solutions, and if the discriminant is <0 there are 2 imaginary (complex) solutions.
Thank you,
MrB