'Solving' a quadratic means finding it's roots - the x values for which the function = 0. i.e. you are solving for x:
You can re-arrage this formula by completing the square to solve for x, which is called the quadratic formula, and looks like:
Now plugging values from a given quadratic into here will give you your two roots, whether tey be real,equal, or complex.
Obviously on agraph of a function, it's roots will be the x values where the function crosses the y-axis (f(x)=0).
Finally, factoring is the opposite of expanding, and is used to easily see the roots of a function in factorised form, by using the factor theorem.