Answer
Total number of teams played is 9.
Work Step by Step
According to given formula,$N=\frac{{{t}^{2}}-t}{2}$.
It can be arranged in the form of quadratic equation as:
$\begin{align}
& 2N={{t}^{2}}-t \\
& {{t}^{2}}-t-2N=0
\end{align}$
The value of $N$ is 36.So, quadratic equation becomes:
$\begin{align}
& {{t}^{2}}-t-2\left( 36 \right)=0 \\
& {{t}^{2}}-t-72=0 \\
& {{t}^{2}}-9t+8t-72=0 \\
& t\left( t-9 \right)+8\left( t-9 \right)=0
\end{align}$
Further calculation shows that:
$\begin{align}
& \left( t+8 \right)\left( t-9 \right)=0 \\
& t=-8,9
\end{align}$
Here, the number of teams cannot be in negative number, so $t=-8$can be neglected.
Therefore, the number of teams played is$t=9$.
The total number of teams can be played in the league with 36 matches is 9.
