The number of intersection points of the graphs of $f(x)=x^2+5x+7$ and $g(x)=x+3$? What is the number of intersection points of the graphs of $f(x)=x^2+5x+7$ and $g(x)=x+3$?
I solved the equation $$x^2+5x+7=x+3\\x^2+4x+4=0\\(x+2)^2=0\\x=-2$$
What does $x=-2$ mean? I am not sure even if I am supposed to solve the equation. How can we think about it? Thank you in advance!

 A: The number of roots of $f(x) = g(x)$ are precisely the number of intersection points of the two curves, $f(x)$ and $g(x)$. In your case, there is a repeated root, namely $x = -2$. So, there  is only one distinct point of intersection.
To better understand why we solve $f(x) = g(x)$ to get the intersection points - ask yourself, what is an intersection point? It is a point that lies on both the curves $f$ and $g$, so it better take the same value on both the curves (or you should see a very visual contradiction).
A: The "$x = -2$" part just represents the $x$-coordinate of the point of intersection between the two graphs. Also, since you only got one $x$-coordinate, this just represents that there is only one intersection point. But for this "$x = -2$" to make sense, you need to plug into one of the original functions to get the $y$-coordinate, where the $y$-coordinate for this intersection point is $f(x)$ or $g(x)$, deciding on whatever function you want to plug the $x$-coordinate back into. In this case, let's say you plug it back into the function $g(x) = x + 3$. Substitute $x = -2$ back into this equation. $g(x) = 1$. So, the $y$-coordinate is $1$ for this intersection point. So the intersection point that you would be visualizing would be the point $(-2, 1)$.
