To find the width of the trampoline, we need to solve the quadratic equation given:
\[ w^2 + 6w - 112 = 0 \]
We can solve this equation using the quadratic formula, which is given by:
\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \(a = 1\), \(b = 6\), and \(c = -112\). Now, we substitute these values into the quadratic formula:
- Calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot (-112) = 36 + 448 = 484 \]
- Now, take the square root of the discriminant:
\[ \sqrt{484} = 22 \]
- Substitute back into the quadratic formula:
\[ w = \frac{-6 \pm 22}{2 \cdot 1} = \frac{-6 \pm 22}{2} \]
This gives us two possible solutions:
\[ w = \frac{16}{2} = 8 \]
or
\[ w = \frac{-28}{2} = -14 \]
Since width cannot be negative, we discard \(w = -14\).
Thus, the width of the trampoline is:
\[ \boxed{8 \text{ ft}} \]
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