Negative Solutions

So, a simple mind like mind would ask the question, if the positive solution of this equation represents the explosive, radiating force, then what does the negative solution to this equation represent?

Wouldnâ€™t the negative solution represent the OPPOSITE of the positive energy solution?

I have read lots of papers where the negative energy solutions are just as valid as the positive energy solutions, where they are interpreted as a strong binding force of some kind, maybe gravity, maybe the strong nuclear force that holds the atom together or maybe both.