Parsing numbers from equations into strings
Without changing things too much...


You need some #includes.
Don't use using namespace. One of the issues is knowing where a function comes from (which one is being used) - this hides it. Using a short std:: gains you a lot in understandability. You can do using std::string which is better, but imo even that isn't worth it.
You might as well put printHelp before main rather than after, that way you don't have to declare it.  And it's not used currently anyway. 
Matrix also isn't used.
Having an Equation class doesn't make sense for what you're (currently) using it for. You might as well just have a naked function.
You don't allow for the possibility that firstx may be invalid, ie there are no x in the equation. This may be a given, of course. And for that matter, that the x^2 term will be first!
If secondx is invalid, skip over processing it.
Multiple breaks in while loops should be avoided, for readability. This (first) loop probably should be split off into another function since it's used twice, together with the following section.
I'm a little nervous of int startloc = firstx - 1;, since firstx is unsigned and startloc isn't. I'd cast firstx to an int first if I were using it like this.
Consider using std::string::find_last_of and std::string::substr. They'll simplify the code somewhat.
Check that you don't lose a negative coefficient.