To find the distance between the edges of the arches defined by the quadratic equation \( y = -x^2 + 2x + 8 \), we need to first determine the x-coordinates where the arches intersect the x-axis (i.e., where \( y = 0 \)). This occurs when:
\[ -x^2 + 2x + 8 = 0 \]
We can rearrange this to:
\[ -x^2 + 2x + 8 = 0 \quad \Rightarrow \quad x^2 - 2x - 8 = 0 \]
Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -2 \), and \( c = -8 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-2)^2 - 4(1)(-8) = 4 + 32 = 36 \]
Now applying the quadratic formula:
\[ x = \frac{-(-2) \pm \sqrt{36}}{2(1)} = \frac{2 \pm 6}{2} \]
This gives us two solutions:
\[ x = \frac{8}{2} = 4 \quad \text{and} \quad x = \frac{-4}{2} = -2 \]
The edges of the arches are at \( x = -2 \) and \( x = 4 \). To find the distance between these points on the x-axis:
\[ \text{Distance} = |4 - (-2)| = |4 + 2| = 6 \]
Thus, the distance between the edges of the arches is \( 6 \) units.
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