Suppose the width of the play area is \(x\) feet. Because there are three sections of fence along the width of the play area, that leaves \(300 - 3x\) feet of fence for its length. The area of the play area should be \(6000\) square feet, so we have the equation
\begin{equation*}
x(300 - 3x) = 6000
\end{equation*}
This is a quadratic equation. In standard form,
\begin{equation*}
\begin{aligned}[t]
3x^2 - 300x + 6000 \amp= 0\amp\amp \blert{\text{Divide each term by 3.}}\\
x^2 - 100x + 2000 \amp= 0
\end{aligned}
\end{equation*}
The left side cannot be factored, so we use the quadratic formula with \(a = \alert{1}\text{,}\) \(b = \alert{-100}\text{,}\) and \(c = \alert{2000}\text{.}\)
\begin{equation*}
\begin{aligned}[t]
x \amp=\frac{-(\alert{-100}) \pm\sqrt{(\alert{-100})^2 - 4(\alert{1})(\alert{2000})}}{2(\alert{1})}\\
\amp= \frac{100 \pm\sqrt{2000}}{2}\approx \frac{100 \pm 44.7}{2}
\end{aligned}
\end{equation*}
Simplifying the last fraction, we find that \(x \approx 72.35\) or \(x\approx 27.65\text{.}\) Both values give solutions to the problem.
If the width of the play area is \(72.35\) feet, then the length is \(300 - 3(72.35)\text{,}\) or \(82.95\) feet.
If the width is \(27.65\) feet, the length is \(300 - 3(27.65)\text{,}\) or \(217.05\) feet.