To solve the equation \((2x + 10)(2x + 25) = 936\), we first expand the left side of the equation:
\[ (2x + 10)(2x + 25) = 4x^2 + 50x + 20x + 250 = 4x^2 + 70x + 250 \]
Now, we set this equal to 936:
\[ 4x^2 + 70x + 250 = 936 \]
Next, we want to set this equation to zero by subtracting 936 from both sides:
\[ 4x^2 + 70x + 250 - 936 = 0 \]
This simplifies to:
\[ 4x^2 + 70x - 686 = 0 \]
To make calculations easier, we can divide the entire equation by 2:
\[ 2x^2 + 35x - 343 = 0 \]
Now, we can use the quadratic formula, given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 35\), and \(c = -343\).
First, we calculate the discriminant:
\[ b^2 - 4ac = 35^2 - 4 \cdot 2 \cdot (-343) = 1225 + 2744 = 3969 \]
Now, we take the square root of the discriminant:
\[ \sqrt{3969} = 63 \]
Now, we can substitute back into the quadratic formula:
\[ x = \frac{-35 \pm 63}{2 \cdot 2} = \frac{-35 \pm 63}{4} \]
This gives us two solutions:
- \(x = \frac{-35 + 63}{4} = \frac{28}{4} = 7\)
- \(x = \frac{-35 - 63}{4} = \frac{-98}{4} = -24.5\)
Now we have our two solutions:
- \(x = 7\) (this is a valid solution since distance cannot be negative)
- \(x = -24.5\) (this does not make sense as a distance)
Filling in the statement:
\(x = 7\) is a solution to the equation, and is the distance between the pool and the fence.
\(x = -24.5\) is a solution to the equation, but it does not make sense as the distance between the pool and the fence.
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