SOLUTION: In a sports league of n teams in which each team plays every other team twice, the total number N of games to be played is given by N = n^2 - n. How many teams are in a softball le

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: In a sports league of n teams in which each team plays every other team twice, the total number N of games to be played is given by N = n^2 - n. How many teams are in a softball le      Log On


   

Question 173198This question is from textbook Introductory Algebra
: In a sports league of n teams in which each team plays every other team twice, the total number N of games to be played is given by N = n^2 - n. How many teams are in a softball league if the total number of games played is 42? This question is from textbook Introductory Algebra

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"if the total number of games played is 42", this means that N=42


N+=+n%5E2+-+n Start with the given equation


42+=+n%5E2+-+n Plug in N=42


0+=+n%5E2+-+n+-+42 Subtract 42 from both sides.


Notice we have a quadratic equation in the form of an%5E2%2Bbn%2Bc where a=1, b=-1, and c=-42


Let's use the quadratic formula to solve for n


n+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


n+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-42%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-1, and c=-42


n+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-42%29+%29%29%2F%282%281%29%29 Negate -1 to get 1.


n+=+%281+%2B-+sqrt%28+1-4%281%29%28-42%29+%29%29%2F%282%281%29%29 Square -1 to get 1.


n+=+%281+%2B-+sqrt%28+1--168+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-42%29 to get -168


n+=+%281+%2B-+sqrt%28+1%2B168+%29%29%2F%282%281%29%29 Rewrite sqrt%281--168%29 as sqrt%281%2B168%29


n+=+%281+%2B-+sqrt%28+169+%29%29%2F%282%281%29%29 Add 1 to 168 to get 169


n+=+%281+%2B-+sqrt%28+169+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


n+=+%281+%2B-+13%29%2F%282%29 Take the square root of 169 to get 13.


n+=+%281+%2B+13%29%2F%282%29 or n+=+%281+-+13%29%2F%282%29 Break up the expression.


n+=+%2814%29%2F%282%29 or n+=++%28-12%29%2F%282%29 Combine like terms.


n+=+7 or n+=+-6 Simplify.


So the possible answers are n+=+7 or n+=+-6


However, you cannot have a negative number of teams. So this means that the only solution is n+=+7



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Answer:


So the solution is n+=+7 which means that there are 7 teams in the league.