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.

In addition to the excellent answers above, we typically bring $0$ on the right hand side just to factorize $ax^2+bx+c$ expression on the left hand side and then use the Zero-product property of real numbers. If you have $ax^2+bx+c=1$ and after factorization you have $a(x-r)(x-s)=1$ then you got no property to use and find the values of $x$.
