Jack's solution to the problem of the beast is to hunt and kill it. He approaches everything from a childish and savage point of view. In his mind, if something is scary or threatening, it needs to be dealt with in as quick a way as possible. He doesn't stop to think if they actually have a chance at killing it; he just leads the boys to search its lair and drive it out. Jack is constantly trying to prove himself as a leader, and his main goal is to be in power. Thus, if he kills the beast, he will destroy Ralph's weak hold on the leadership position.
Jack has no set plan for his idea. He doesn't stop to think things through, or develop a concrete blueprint for dealing with the beast. Instead, he rounds up the rest of the boys and marches off to fight, without knowing what the beast may be, or if they're capable of killing it. It is not the most mature way of dealing with the issue, nor is it the most rational. His response represents the more savage qualities in every human, lying just below the surface of rational though.