What does a subwoofer solve?

Subs are meant for low freq extension. They "solve" nothing on their own.

Subs will make your entire system run better. They will lower the distortion in your mains when using a highpass filter. They will give your system more dynamic range (output). They will most likely add bass extension depending on your speaker. Subs done right are always better than no subs. 
 

not to mention you can place the subs for bass while not effecting the soundstage negatively. I can’t recommend subs enough but and a big but is you need a highpass filter to sound right. 

Your loudspeakers are -6db at 28 hz. I can't guess the 3db point but a good sub will be flat into the low 20s. There still is some information down there that is worth exploring. 

do you need a subwoofer ifd your source does'nt have the lows that a subwoofer would produce??

g_nakamoto

  I think that your question answers itself. 

 In my case, I was appreciative of the difference in performance that a good sub made. In a few recordings, I knew that the subsonic signature was there, but didn't experience it until the was in place. Yes, partially due to the fact that main speakers were not burdened with the task, but also because the information was there and is now being reproduced to a higher degree.