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, 55 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: Substitute N=55 into formula

$$55 = \frac{t^2 - t}{2}$$

Step2: Eliminate denominator (multiply by 2)

$$110 = t^2 - t$$

Step3: Rearrange to quadratic form

$$t^2 - t - 110 = 0$$

Step4: Factor the quadratic equation

$$(t - 11)(t + 10) = 0$$

Step5: Solve for valid t (positive)

$t - 11 = 0 \implies t=11$; discard $t=-10$ (invalid, no negative players)

Answer:

11 players