v=0 to find the endpoints of displacement. For this you need v=f(x) so you need to use the speciel result ( a=0.5v^2).
Centre of motion is simply the two endpoints, divide by 2, so (x1+x2)/2
I think the centre of motion would be when the particle has a displacement of 0, so you would let x=0 to find the time(s) that this occurs. Correct me if I'm wrong.