If x=6 is the only x-intercept of the graph of a quadratic equation, which statement best describes the discriminant of the equation?

If x=6 is the only intercept, it means that the quadratic is tangential to the x-axis, or equivalently x=6 is a root with multiplicity 2.
The discriminant is typically zero for multiplicity 2 in a quadratic. 
It will be clear looking at the quadratic formula:
if D=b^2-4ac = discriminant of the quadratic, then
x1=(-b+sqrt(D))/2a or x2=(-b-sqrt(D))/2a
if x1=x2 (double root, or multiplicity 2) then clearly D has to be zero.