Remember that the solutions, or zeroes, or roots of a function are the points where the function outputs zero. In other words, if [tex] f(x^*) = 0 [/tex], then [tex] x^* [/tex] is a solution/zero/root of [tex] f(x) [/tex]. If this is the case, then the function can be factored as the multiplication of parenthesis like [tex] (x-x^*) [/tex].
As a graphical consequences, the zeroes of a function correspond to the points [tex] x = x^*,\ y = 0 [/tex], and all points like [tex] (x^*,0) [/tex] are on the x-axis.
So, the graph of a function touches the x axis at [tex] x^* [/tex] if and only if [tex] x^*[/tex] is a solution for the function.
Now, your graph represents a parabola. A parabola can either have:
- No solutions, meaning that it never touches the x axis
- Two coincident solutions, meaning that it touches the x axis "twice", in the same point
- Two different solutions, meaning that it crosses the x axis in two different points.
You parabola has two coincident solutions, because it only touches the x axis twice at [tex] x = 5 [/tex].
This means that [tex] x = 5 [/tex] is twice a solution of the parabola, and as such, it implies the factorization [tex] (x-5)(x-5) = (x-5)^2 [/tex]
