First of all, sorry for not using the template, but I think in this situation it's better to explain my problem right away:
I'm studying for a physics test, but I think I don't really understand Galilean invariance. In my textbook there is an example in which they proof that if you consider 2 frames S and S' in standard configuration that the second law of Newton is Galilean invariant by proofing that if [itex]x' = x - Vt[/itex] than [itex]F_x = F'_x[/itex], so this law holds in both frames. So far I understand this.

However, in the book there is one assignment in which they ask me to verify that the relationship between kinetic energy and momentum, [itex]E = p^2/2m[/itex], is Galilean invariant. I couldn't really figure it out by myself so I looked at the answers. The answer is as followed:

Source: McComb, W. D., 1999. Dynamics and Relativity. New York: Oxford University Press Inc.

I understand all of the equations, however I just don't understand why this verifies that this relationship is Galilean invariant.

You can see that this expression for E' is the same as what is obtained if you assume that E' = (p')^{2}/2m, showing that this relation between kinetic energy and momentum does indeed hold in S' as well.