How many points of intersection do
the two graphs have?
We name the graphs A and B.
Let the eqation of A be f(x,y)=0, and
B g(x,y)=0.
The points of intersection of A and B
are the roots of simultaneous equations,
f(x,y)=0 and g(x,y)=0.
Conversely, the roots are the points of
intersection of A and B.
The reason is that both are
{ (x,y) | f(x,y)=0, g(x,y)=0 }.
So all you have to do in order to seek
the number of points of intersection is
to count roots of the equations.