To solve the equation \( 64x^2 + 9 = 13 \) by factoring using the difference of squares pattern, we first rearrange the equation:
\[ 64x^2 + 9 - 13 = 0 \implies 64x^2 - 4 = 0 \]
Next, we can factor this equation. Notice that \( 64x^2 \) and \( 4 \) can be expressed as perfect squares:
\[ (8x)^2 - 2^2 = 0 \]
This is now a difference of squares, which can be factored as:
\[ (8x - 2)(8x + 2) = 0 \]
Setting each factor equal to zero gives us:
- \( 8x - 2 = 0 \)
- \( 8x + 2 = 0 \)
For the first equation:
\[ 8x - 2 = 0 \implies 8x = 2 \implies x = \frac{2}{8} = \frac{1}{4} \]
For the second equation:
\[ 8x + 2 = 0 \implies 8x = -2 \implies x = \frac{-2}{8} = -\frac{1}{4} \]
So the solutions to the equation \( 64x^2 + 9 = 13 \) are:
\[ x = \frac{1}{4}, \quad x = -\frac{1}{4} \]
The correct solutions among the given options are:
x equals negative Start Fraction 1 over 4 End Fraction, x equals Start Fraction 1 over 4 End Fraction.