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? (square) players (simplify your answer.)

Explanation:

Step1: Substitute N with 45

We know the formula \( N=\frac{t^{2}-t}{2} \), and \( N = 45 \). So we substitute \( N \) into the formula:
\( 45=\frac{t^{2}-t}{2} \)

Step2: Multiply both sides by 2

To eliminate the denominator, we multiply both sides of the equation by 2:
\( 45\times2=t^{2}-t \)
\( 90=t^{2}-t \)

Step3: Rearrange into quadratic equation

Rearrange the equation to the standard quadratic form \( ax^{2}+bx + c = 0 \). Here, we get:
\( t^{2}-t - 90=0 \)

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 equation as:
\( (t - 10)(t+9)=0 \)

Step5: Solve for t

Set each factor equal to zero and solve for \( t \):

• For \( t - 10=0 \), we get \( t = 10 \)
• For \( t + 9=0 \), we get \( t=-9 \)

Since the number of players \( t \) cannot be negative, we discard \( t=-9 \).

Answer:

10