I was one of those kids. I never had a stable home, I have 5 siblings with many parents, I moved between different parents, lived on the floor with mom and siblings at one of her friends homes for a few months etc. But mustering up the energy to put in effort for one test is still possible, doing that for everything, no way.
Kids aren't stupid, they have spent years doing things, reading things, it isn't hard to find out information about college, SAT etc. A bad home doesn't mean you are stupid even though you make it sound like it does.