Question: Typically, when you look at a real world examples for using a quadratic equation/formula, you get a ball being tossed or a missile being launched. I understand what each component of the equation stands for. However, my question is what angle is the object being thrown? I would think that the angle of trajectory would change the height and distance the object reaches. Nevertheless, all the real world examples never talk about this and use the same set up no matter if its a ball or missile being thrown. As such, is there a typical/average angle these real world examples are using?

Answer: 
Your equation only applies to the altitude (y-coordinate) of the object. You need a separate equation for the horizontal component (x-coordinate). With both equations, you can figure out the angle. If $x$ and $y$ are functions of time, $y=y(t)$ and $x = x(t)$, then the angle is the inverse tangent of
$$ \dfrac {dy}{dx} = \dfrac{dy/dt}{dx/dt}$$
Typically, such a problem will state a particular initial angle at which the projectile is thrown. Sometimes a problem will say "vertically", so the angle with the horizontal is 90 degrees, sometimes "horizontally" so the angle is 0 degrees.