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.

It is by definition. The x's that make the entire equation equal 0 are called the "solutions" to the equation. In other words, the x coordinates of the  points where the graph intersects the x axis would be the solutions or otherwise called zeros.
