11y=ax2+bx+c
y=a[(x-b/2a)2+4ac-b2/4a2]
from the fact that when a>0 the graph renders a parabola with upward concavity.
secondly when d<0 it suggests that graph never cuts the x axis
which implies that the graph will lie compleely above the x axis and thus for any value of x on the number line the value of the expression will be positive.
now when the latter case comes a,0 which implies the parabola points downward and thus on similar grounds you say that the graph lies completely below the x axis.
hope youve understood.
you are free to ask any question on this topic