Solution forms for the CFs of 2nd order differential equations: how are they derived?

I'm currently in year 12 studing FP2 maths and we've just finished second order differential equations. One of the things that bugs me about Edexcel is their lack of proof (I know some things are out of spec, but a pointer to a site or something would be good); more particularly, for the deriving of the results for solution forms for complementary functions, for example:

"If the auxilliary equation has two distinct real roots, p and q, then the solutions is of the form y = Ae^(px) + Be^(qx)" - why must we add them? If you check each term on their own, they both satisfy the original homogeneous differential equation. It is little things like this that bug me, so I'm wondering:

Is there a good explanation for how we ARRIVE at these results? I would just be curious to know is all :)

Many thanks