To find a point of intersection between 2 functions, why is the solution to make each function equal the other?

Question

If $f(x)=x^2−6x+14$ and $g(x)=−x^2−20x−k$, determine the value of $k$ so that there is exactly one point of intersection between the two parabolas.

I want to clarify that I know how to solve this problem.

My question: why does making both function equal each other allow us to determine that point of intersection? I am interested in the proof if anyone can point me in the right direction.

Answer 1

hint

A point of first parabola has coordinates $$(x,f(x))$$ a point of the second parabola is of the form $$(x,g(x))$$

the point of the intersection must satisfy $$(x,f(x))=(x,g(x))$$ thus $$f(x)=g(x).$$

Answer 2

Hint:

$$f(x)=g(x)\implies f(x)-g(x)=0$$

So set $h(x)=f(x)-g(x)=2x^2+14x+(14+k)=0$

and use the discriminant to find when this has only one solution.