Prove that s^2=(s')^2 using the Lorentz Transformation

1. The problem statement, all variables and given/known data I am learning special relativity and we came across the invariant quantity s = x² - (ct)². Our professor wants us to prove it. I admit that this is a proof and belongs in the mathematics section but I didn't see an Algebra section and this is most easily identified by those learning special relativity.



The assignment simply states



"Prove s² = s'²"





2. Relevant equations

s²= x²-(ct)²



[itex]\gamma[/itex]=[1-([itex]\frac{v}{c}[/itex])²]^-1/2



x' = [itex]\gamma[/itex](x-vt)



t' = [itex]\gamma[/itex](t-(vx/c²)



3. The attempt at a solution



My textbook is telling me in one sentence that if we apply the lorentz transformation to x and t then s² = s'².....so I did that...



I choose to start with s'² = x'²-(ct')²



Applying the lorentz transformation to x' and t' our equation becomes...



s'² = ([itex]\gamma[/itex](x-vt))²-(c([itex]\gamma[/itex](t-(vx/c²))²



Expanding what we have takes us to...



s'² = ([itex]\gamma[/itex]²(x²-2vt+(vt)²)-(c²[itex]\gamma[/itex]²(t²)-2(v/c²)x+(v²/c⁴)x²))



If I combine some terms...



s'² = [itex]\gamma[/itex]²[x²(1-(v²/c⁴)+t²(v-1)+2v((x/c²)-t)]



From here I tried a couple of different things on scratch paper but I couldn't see particular direction that would simplify it all down. Am I just not being patient enough and not seeing that it gets worse before it gets better?



Thanks in advance.