If your graph hits the x-axis at only one point, then the discriminant is zero. This only occurs because we can ignore the "plus or minus" part of the quadratic formula if the discriminant is zero, giving what is called a "double-root" to the equation (that means one answer, occurring twice).
If the discriminant is >0 you will have two real, distinct roots. If this was the case with your graph you would have two points of intersection with the x-axis.
If the discriminant is <0 you will have two imaginary, distinct roots, in which case you could not graph the points of intersection on the real plane at all, so the graph will have no intersection with the x-axis.