In a round-robin tournament with $n$ teams, the number of games that must be played is $(n^2-n)/2$. How many teams are in a round-robin tournament in which 55 games are played?
I need to find the value of $n$ that satisfies the equation $(n^2-n)/2=55$. I can multiply both sides by 2 to get rid of the fraction: $n^2-n=110$. I can rearrange the terms to make a quadratic equation: $n^2-n-110=0$. I can factor the quadratic equation into two binomials: $(n-11)(n+10)=0$. The solutions are $n=11$ and $n=-10$. However, $n=-10$ does not make sense in the context of the problem, since there cannot be a negative number of teams. Therefore, the only valid solution is $n=11$. This means there are 11 teams in the tournament. 