Problem 131 - Algebra and Trigonometry

Question

In a round-robin chess tournament, each player is paired with every other player once. The formula $$ N=\frac{x^{2}-x}{2} $$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve. In a round-robin chess tournament, 21 games were played. How many players were entered in the tournament?

Step-by-Step Solution

Answer

There were 7 players in the tournament.

Step 1: Substitute the given number of games into the equation

Replace \(N\) in the equation with 21, getting \[21 = \frac{x^{2}-x}{2}\]

Step 2: Simplify the Equation

Multiply both sides of the equation by 2 to eliminate the fraction, resulting in \[42=x^{2}-x\]. Then rewrite the equation in standard quadratic form \[0 = x^2 - x - 42\].

Step 3: Factor the quadratic equation

Now we have to factor the quadratic equation. \[(x - 7)(x + 6) = 0\] will be our factored form.

Step 4: Solve for \(x\)

Setting each factor equal to zero gives the solutions \(x - 7 = 0\) and \(x + 6 = 0\). Adding \(7\) to both sides of the first equation gives \(x = 7\). Subtracting \(6\) from both sides of the second equation gives \(x = -6\).

Step 5: Reject the non-sensible solution

Since we can't have negative players, \(x = -6\) doesn't make sense in this context. So there were \(7\) players in the tournament.

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical statement involving a variable raised to the second power, typically in the form of \( ax^2 + bx + c = 0 \). In our example from the tournament problem, the quadratic equation is \( x^2 - x - 42 = 0 \). Quadratic equations often appear in scenarios involving paths, areas, and as we see here, when modeling the number of matches in a round-robin tournament.
Solving quadratic equations requires us to find the value(s) of \( x \) that satisfy the equation. This means we are looking for numbers that make the whole expression equal to zero. Understanding quadratic equations is an important skill as they frequently show up in various real-world applications, including physics and economics. In this case, it helps us determine how many players took part based on the number of games played.

Factoring

Factoring is a method used to simplify a quadratic equation. It involves expressing the equation as a product of two smaller equations, also known as factors. In the case of \( x^2 - x - 42 = 0 \), we need to find two numbers that multiply to -42 (the constant term) and add up to -1 (the coefficient of \( x \)).
For this quadratic, those numbers are -7 and 6. We express the quadratic equation as the product of these two factors: \((x - 7)(x + 6) = 0\).

• This method works well when numbers are suitable for easy factoring.
• It allows us to break down complex expressions into simpler components, making it easier to find solutions for \( x \).

Learning to factor efficiently is key to solving many algebra problems quickly.

Substitution Method

The substitution method is a technique often used in mathematics to simplify expressions or equations by replacing one element with another. In the context of this exercise, substitution refers to the step where we substitute the given value for \( N \) (the number of games) into the round-robin formula. We replace \( N \) with 21, resulting in the equation \( 21 = \frac{x^{2}-x}{2} \).
After substituting and simplifying the equation by multiplying through by 2, we directly relate the problem requirements to the number of players, \( x \). This method transforms the known into a solvable expression for the unknown, making the problem easier to handle. By substituting the known value, we create a clearer path to find \( x \), the number of players.

Number of Players

The number of players, represented by \( x \), is what we are ultimately trying to find in the context of a round-robin tournament problem. These are participants who each play against every other player exactly once. In mathematical terms, the count of these players relates directly to the number of games played.
Theoretically, if there are more players, the number of possible games increases drastically. That is why this situation is modeled using a quadratic expression, which effectively handles combinations. By solving the quadratic equation and arriving at only the positive solution, as the number of players can't be negative, we find that the tournament had 7 players. Understanding this link between the solution of the quadratic equation and the context aids in cementing knowledge about how math can be applied to solve real-life situations, like organizing a tournament.