To solve the equation \( 64x^2 + 9 = 13 \) by factoring using the difference of squares pattern, we first need to rearrange the equation in a standard form.
Start by moving the constant term on the right side of the equation to the left side:
\[ 64x^2 + 9 - 13 = 0 \] which simplifies to: \[ 64x^2 - 4 = 0 \]
Now, we can factor this expression. Notice that we can rewrite \( 4 \) as \( 2^2 \):
\[ 64x^2 - 4 = (8x)^2 - (2)^2 \]
This is a difference of squares, which can be factored as follows:
\[ (a^2 - b^2) = (a - b)(a + b) \]
Here, \( a = 8x \) and \( b = 2 \). Thus we can factor it into:
\[ (8x - 2)(8x + 2) = 0 \]
Next, we set each factor equal to zero to find the solutions:
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\( 8x - 2 = 0 \) \[ 8x = 2 \quad \Rightarrow \quad x = \frac{2}{8} = \frac{1}{4} \]
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\( 8x + 2 = 0 \) \[ 8x = -2 \quad \Rightarrow \quad x = \frac{-2}{8} = -\frac{1}{4} \]
Thus, the solutions to the equation \( 64x^2 + 9 = 13 \) are:
\[ x = \frac{1}{4} \quad \text{and} \quad x = -\frac{1}{4}. \]