To solve the equation \(64x^2 + 9 = 13\) by factoring using the difference of squares pattern, we first rearrange the equation to set it to zero:
\[ 64x^2 + 9 - 13 = 0 \implies 64x^2 - 4 = 0 \]
Next, we can rewrite \(64x^2 - 4\) to fit the difference of squares format:
\[ (8x)^2 - (2)^2 = 0 \]
Now we can factor this expression:
\[ (8x - 2)(8x + 2) = 0 \]
Setting each factor equal to zero gives us the possible solutions:
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\(8x - 2 = 0\)
- Solving for \(x\): \[ 8x = 2 \implies x = \frac{2}{8} = \frac{1}{4} \]
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\(8x + 2 = 0\)
- Solving for \(x\): \[ 8x = -2 \implies x = \frac{-2}{8} = -\frac{1}{4} \]
Thus the solutions to the equation are:
\[ x = \frac{1}{4} \text{ and } x = -\frac{1}{4} \]
From the options provided, the correct solutions are:
- \(x = -\frac{1}{4}, x = 14\)
However, it appears that the value \(x=14\) is not derived from our calculations. Only \(x = \frac{1}{4}\) and \(x = -\frac{1}{4}\) are valid solutions.
Revisiting that option looks like there might be a small oversight in relaying the answer format in options.
Thus the accurate extracted solutions for the starting equation are:
- \(x = -\frac{1}{4}\)
- \(x = \frac{1}{4}\)
Please check the answer choices once more as they might need refinement based on our calculations of obtained solutions.
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