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In a sports league of x teams in which all teams play each other twice, the total number $N$ of games played is given by$$x^{2}-x=N$$The teams in a women's softball league play each other twice, for a total of 132 games. How many teams are in the league?

12

Algebra

Chapter 6

Polynomial Factorizations and Equations

Section 7

Applications of Polynomial Equations

Graphs and Statistics

Exponents and Polynomials

Equations and Inequalities

Polynomials

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Lectures

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Oh, right. So the question we're gonna be looking at today is an algebra problem that involves trying to find a mate. Teams are in a given sports league. So this question starts out by saying, um, a league has an end number of teams on any number of teams where each team plays each other twice, and upper case n equals the number of games played. We're getting that from the problem. Um, and this is given by the equation and equals and squared or sorry. And the question they stated his ex X squared minus X. So the X squared and earned the axe in general represents the number of teams in the league. Um, so the first thing we're given in this problem is that the teams play 132 games. So right off the bat, we're gonna write an equals 32. So 1 32 equals X squared minus X. First thing we're gonna do after that is, um, bringing this to the other side and make it zero. So zero is equal to X squared minus X minus 1 32 The next up we're gonna do is we're going to factor it just to see, um to attempt to get are two possible answers. Answers. For this question, you can either use quadratic formula or trying to find the number that multiplies to 1 32 and adds to negative one. Um, when you do this, either way you're going to be left with, um, you're gonna be left with zero equals X plus 11 and X minus 12. These are your two factors for this question. You're going Teoh. See what the equal and you're gonna be left with. X equals negative 11 and X equals 12 For these types of world problems, you're gonna have to use a bit of logic at the end to figure out your answer. Um, obviously, in a sports league, you can't have negative answers. You can have negative amount of teams. Um, so in this case, the possible answers we have our negative 11 and 12 you can have a night, even member teams. So that means that X equals 12 has to be your answer. So therefore, there are 12 tears in the league, and that is your final answer.

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