Explain why a line and a parabola can have no more the two points of intersection.

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. 

Where’s the