To determine the number of players in a round-robin tennis tournament where each player plays every other player exactly once and 120 matches were played, we start with the given relationship:
[tex]\[ N = \frac{1}{2} n(n-1) \][/tex]
Here, [tex]\(N = 120\)[/tex]. We need to solve for the number of players, [tex]\(n\)[/tex].
1. Begin by substituting the given number of matches into the equation:
[tex]\[ 120 = \frac{1}{2} n(n-1) \][/tex]
2. To eliminate the fraction, multiply both sides of the equation by 2:
[tex]\[ 2 \times 120 = n(n-1) \][/tex]
[tex]\[ 240 = n^2 - n \][/tex]
3. Rearrange the equation to form a standard quadratic equation:
[tex]\[ n^2 - n - 240 = 0 \][/tex]
This is now a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -240 \)[/tex].
4. Solve the quadratic equation using the quadratic formula:
[tex]\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-240)}}{2 \cdot 1} \][/tex]
[tex]\[ n = \frac{1 \pm \sqrt{1 + 960}}{2} \][/tex]
[tex]\[ n = \frac{1 \pm \sqrt{961}}{2} \][/tex]
[tex]\[ n = \frac{1 \pm 31}{2} \][/tex]
5. This yields two potential solutions:
[tex]\[ n = \frac{1 + 31}{2} = \frac{32}{2} = 16 \][/tex]
[tex]\[ n = \frac{1 - 31}{2} = \frac{-30}{2} = -15 \][/tex]
Since the number of players [tex]\(n\)[/tex] must be a positive integer, we discard the negative solution, [tex]\(n = -15\)[/tex].
6. Thus, the number of players in the tournament is:
[tex]\[ n = 16 \][/tex]
