What conditions give 2 roots, one root and no root?

The discriminant D is found by this formula

D = b^2 - 4ac

there are 3 possible outcomes

1) If D is smaller than 0, ie D < 0, then there are no roots

2) If D is equal to zero (D = 0), then there is exactly one root. This root is rational assuming the coefficients are integers. This is called a double root

3) If D > 0, then there are 2 different roots. If D is a perfect square, then these two roots are rational numbers (aka fractions of whole numbers) assuming the coefficients are integers. If D is not a perfect square, then the two roots are irrational numbers.