Martin Luther King changed the world because America decided that it would rather deal with him, then the far more violent parts of the black liberation movement.
If those more violent parts didn't put the fear of God into the government, he could have been safely ignored.
Responding to political concerns with violence just begets more violence. It just entrenches people's beliefs even further, inspires revenge from the 'other side' (or at least, their own axe crazy extremists) and eventually causes everything to erupt even further (through a civil war).
Honestly, I'd say this part in the most recent series of Doctor Who sums up some of the issues with people trying to solve all their problems with violence:
https://www.youtube.com/watch?v=zvGND1i6Dj0