Why are roots (x-intercept) called the solutions of a quadratic equation? As I understand it, all the points in the hyperbole are the solutions. Although I see that it's easier to draw the curve after you discovered the roots, I don't see why to call them 'the solutions.' It has infinite solutions.

The solutions of a quadratic $ax^2+bx+c=0$ are the values of $x$ that make the aforementioned equation true. Even if you had $ax^2+bx+c=1$, you could just subtract $1$ from both sides and that would be your new quadratic. So, they are called solutions of the quadratic because when you substitute the zeros into the quadratic equation, the equation will be satisfied.
Example: $(x+1)(x-1)=0$ has solutions $x=1$ or $x=-1$. This is because those are the only $2$ numbers that would make the equation true.
