Answer
$7$.
Work Step by Step
The function modelling the number of chess games is
$f(x)=\frac{x^2-x}{2}$
where $x$ is the number of chess players.
Plug the given value $f(x)=21$ into the function.
$21=\frac{x^2-x}{2}$
Multiply both sides by $2$ to clear fractions.
$2\cdot 21=2\cdot \left (\frac{x^2-x}{2} \right )$
Clear the parentheses and simplify.
$42=x^2-x$
Subtract $42$ from both sides.
$42-42=x^2-x-42$
Add like terms.
$0=x^2-x-42$
Write the middle term $-x$ as $-7x+6x$
$0=x^2-7x+6x-42$
Group terms.
$0=(x^2-7x)+(6x-42)$
Factor from each terms.
$0=x(x-7)+6(x-7)$
Factor out $(x-7)$
$0=(x-7)(x+6)$
Set each factor equal to zero.
$x-7=0$ or $x+6=0$
Isolate $x$.
$x=7$ or $x=-6$.
Take the positive value because the number of players cannot be negative.
Hence, the number of players who entered in the tournament $=7$.
