Question

in a round - robin chess tournament, each player is paired with every other player once. the function, shown below, models the number of chess games, n, that must be played in a round - robin tournament with t chess players. in a round - robin chess tournament, 45 games were played. how many players entered the tournament?
n=\frac{t^{2}-t}{2}
how many players entered the tournament?
\boxed{} players (simplify your answer.)

Explanation:

Step1: Set up the equation

We know that the number of games \( N \) is given by the formula \( N=\frac{t^{2}-t}{2} \), and we are given that \( N = 45 \). So we set up the equation:
\[
\frac{t^{2}-t}{2}=45
\]

Step2: Multiply both sides by 2

To eliminate the denominator, we multiply both sides of the equation by 2:
\[
t^{2}-t = 90
\]

Step3: Rearrange into standard quadratic form

Subtract 90 from both sides to get the quadratic equation in standard form \( ax^{2}+bx + c = 0 \):
\[
t^{2}-t - 90=0
\]

Step4: Factor the quadratic equation

We need to find two numbers that multiply to -90 and add up to -1. The numbers are -10 and 9. So we can factor the quadratic as:
\[
(t - 10)(t + 9)=0
\]

Step5: Solve for t

Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So we have two equations:
\( t - 10=0 \) or \( t + 9=0 \)
Solving \( t - 10=0 \) gives \( t = 10 \). Solving \( t + 9=0 \) gives \( t=-9 \). But the number of players \( t \) cannot be negative, so we discard \( t=-9 \).

Answer:

10