The discriminant of a function is the b² - 4ac in the quadratic formula.
If it is positive, there are two real solutions. The parabola only has one real solution on the graph (its vertex), so it can't be positive.
If the discriminant is negative, there are no real solutions. In the quadratic formula, you have to take the square root of the discriminant, and you can't take the square root of a negative number to result in real solutions. This doesn't apply to this graph either.
If the discriminant is zero, there is one real solution. You can see the function touching the x-axis in one place, so your answer should be A.