Tournament 61544

In an amateur chess tournament, everyone plays with everyone. There are a total of 171 chess games on the program. How many players take part in the match?

Correct answer:

n =  19

Step-by-step explanation:

$$\begin{gathered} \left(\genfrac{}{}{0ex}{}{n}{2}\right)=\frac{n\cdot (n-1)}{2}=171 \\ n\cdot (n-1)=2\cdot 171 \\ n\cdot (n-1)=2\cdot 171 \\ {n}^2-n-342=0 \\ a=1;b=-1;c=-342 \\ D=b^2-4ac=1^2-4\cdot 1\cdot (-342)=1369 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{1\pm \sqrt{1369}}{2} \\ {n}_{1,2}=\frac{1\pm 37}{2} \\ {n}_{1,2}=0.5\pm 18.5 \\ {n}_1=19 \\ {n}_2=-18 \\ n>0 \\ n=n_1=19 \end{gathered}$$

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