How to prove the number of common points of a tangent and a parabola?

Question

Mathematics

There is a tangent to the graph in the form y = ax^2 + bx + c (parabola).
How to prove that the tangent to any point of the parabola is the only point of intersection with the graph?
It's intuitive, but I don't know how to prove it mathematically

Answer 1

let the tangent at the point (x0, ax0^2+bx0+c);
tangent equation: y - ax0^2+bx0+c = (2ax0+b)(x-x0)
it is necessary to prove that the system // y = ax^2 + bx + c and y - ax0^2+bx0+c = ( 2ax0+b)(x-x0) has units. solution (x0, ax0^2+bx0+c)
we substitute y into the second equation, we give similar ones: x^2-2xx0+x0^2=0 i.e. x=x0 unit solution