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, 1010 games were played. How many players entered the​ tournament?
Upper N equals StartFraction t squared minus t Over 2 EndFraction

The formula given for the number of games played in a round-robin chess tournament is:

\[
N = \frac{t^2 - t}{2}
\]

Where:
- \( N \) is the number of games played,
- \( t \) is the number of players in the tournament.

We know that 1010 games were played, so we can set \( N = 1010 \) and solve for \( t \):

\[
1010 = \frac{t^2 - t}{2}
\]

### Step 1: Multiply both sides by 2 to eliminate the fraction:
\[
2020 = t^2 - t
\]

### Step 2: Rearrange the equation:
\[
t^2 - t - 2020 = 0
\]

Now, we have a quadratic equation of the form \( t^2 - t - 2020 = 0 \).

### Step 3: Solve the quadratic equation using the quadratic formula:
The quadratic formula is:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( t^2 - t - 2020 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -2020 \)

Substitute these values into the quadratic formula:

\[
t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2020)}}{2(1)}
\]
\[
t = \frac{1 \pm \sqrt{1 + 8080}}{2}
\]
\[
t = \frac{1 \pm \sqrt{8081}}{2}
\]

### Step 4: Approximate \( \sqrt{8081} \):
\[
\sqrt{8081} \approx 89.9
\]

### Step 5: Solve for \( t \):
\[
t = \frac{1 + 89.9}{2} \approx \frac{90.9}{2} = 45.45
\]

Since \( t \) must be an integer, we round this to \( t = 45 \).

### Conclusion:
There were **45 players** in the tournament.

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the quadratic formula derived?
2. What would happen if only 1000 games were played instead of 1010?
3. How many games are played with 10 players in a round-robin tournament?
4. Can a non-integer number of games ever occur in a round-robin format?
5. How can you check if your answer for \( t \) is correct?

**Tip:** In a round-robin tournament, each player competes once with every other player, so the number of games grows rapidly with more players.

In a round-robin chess tournament, 1010 games were played. How many players entered the tournament?

Discover how to calculate the number of players in a round-robin chess tournament where 1010 games were played using the quadratic formula. This problem involves algebraic concepts and quadratic equations, suitable for high school students in Grades 9-12.

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