Problem 131 - College Algebra Essentials

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 Exercises \(131-132\). In a round-robin chess tournament, 21 games were played. How many players were entered in the tournament?

Step-by-Step Solution

Answer

The number of players that entered the tournament is 7.

Step 1: Substitute the given value of N into the equation

Substitute \(N = 21\) into the equation \(N = \frac{x^2 - x}{2}\) which yields \(21 = \frac{x^2 - x}{2}\). To simplify the equation, multiply both sides by 2. Doing so, we get \(2*21 = x^2 - x\), which simplifies further into \(42 = x^2 - x\).

Step 2: Transform the equation into standard form for quadratic equation

Getting the equation into the standard quadratic form \(ax^2 + bx + c = 0\), subtract 42 from both sides to get the equation \(x^2 - x - 42 = 0\).

Step 3: Solve for x

The equation \(x^2 - x - 42 = 0\) can be factored into \((x-7)(x+6) = 0\). Therefore, the solutions are \(x = 7\) and \(x = -6\). However, since the number of players cannot be negative, we dismiss \(x = -6\) and accept \(x = 7\) as the number of players.

Key Concepts

Round-Robin Tournament

In the context of chess tournaments, a round-robin format is a system where each player competes against every other player exactly once. It's a popular choice for tournaments because it ensures that all participants have equal opportunities to compete.
This type of tournament provides a comprehensive way to determine the best player by having matches equally distributed.

• Each player competes with every other player.
• Provides fair and extensive coverage of all player matchups.
• Best used when the number of players isn't too large, as the number of matches increases significantly with more participants.

Understanding the structure can help in calculating the number of games played. This leads us to mathematical modeling using quadratic equations to find successful predictions and outcomes.

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations, which are polynomials of the form \(ax^2 + bx + c = 0\).
In our exercise, we transformed the equation \(x^2 - x - 42 = 0\) into a factored form, making it easier to solve for \(x\).

• Find two numbers that multiply to \(c\) (in this case, \(-42\)) and add to \(b\) (in this case, \(-1\)).
• For \(x^2 - x - 42 = 0\), these numbers are \(6\) and \(-7\).
• Rewrite it as \((x - 7)(x + 6) = 0\).
• The solutions are where each factor equals zero: \(x = 7\) and \(x = -6\).

Since the number of players can't be negative, we determine that the number of players is \(7\). This method relies on identifying the correct factors to break the equation down into simpler terms efficiently.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario to predict or understand outcomes.
In this exercise, the formula \(N = \frac{x^2 - x}{2}\) was used to model the number of games played in a round-robin tournament.

• Models help visualize complex scenarios in simpler terms—like understanding tournament structure.
• The given quadratic model accounts for how each additional player increases matches exponentially.
• By substituting known values, predictions about unknowns can be made.
• This exercise precisely demonstrates finding the number of players by inserting the given number of games into the formula.

Mathematical models such as this are powerful tools in predicting and deciphering outcomes in many scientific and practical fields.

Number of Players

Understanding the number of players in a tournament involves unraveling the quadratic equation derived from the modeling formula.
When you know the total games played (\(N\)), you can determine the number of players (\(x\)) by solving the equation \(N = \frac{x^2 - x}{2}\).

• First step is substituting \(N = 21\) into the formula.
• Then, multiply through by \(2\) to remove the fraction: \(42 = x^2 - x\).
• Bring to standard form: \(x^2 - x - 42 = 0\).
• Factor to find possible \(x\) values which indicate the number of players, remembering to discard non-sensible results like negative numbers.

The logic here allows deciphering how a dynamic variable (\(x\), players) interacts within a fixed outcome (\(N\), games played), thus offering insight into managing and planning tournaments efficiently.