A parabola can be written with parameters:
f(x)= ax^2 + bx + c
A line can be written
f(x) = dx + e
where a, b, c, d, e are real numbers.
Now when they intersect the two functions equal eachother.
So:
ax^2+bx+c = dx +e
rearrange:
ax^2 +(b-d)x +c-e=0
Lets call b-d=f and c-e=g, (f, g are still real numbers)
We get:
ax^2+fx+g=0
Now this is the standard form of 2nd degree polynomial. And we know that this can have 0,1,2 real solutions.
Thus we have proven that they cannot have more than 2 points of intersection.
