Answer
The total number teams can be played in the league with 45 matches is 10.
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:
$\begin{align}
& N=\frac{{{t}^{2}}-t}{2} \\
& 2N={{t}^{2}}-t \\
& {{t}^{2}}-t-2N=0
\end{align}$
Substituting the value of $N=45$in above equation as:
$\begin{align}
& {{t}^{2}}-t-2\left( 45 \right)=0 \\
& {{t}^{2}}-t-90=0 \\
& {{t}^{2}}-10t+9t-90=0 \\
& t\left( t-10 \right)+9\left( t-10 \right)=0
\end{align}$
Further calculation shows that,
\[\begin{align}
& \left( t-10 \right)\left( t+9 \right)=0 \\
& t=-9,10
\end{align}\]
Here, the number of teams cannot be in negative number, so $t=-9$can be neglected.
Therefore, the number of teams played is $t=10$.
The total number teams can be played in the league with 45 matches is 10.
