To solve the equation \( 64x^2 + 9 = 13 \) by factoring using the difference of squares, we first rearrange the equation:
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Subtract \( 13 \) from both sides: \[ 64x^2 + 9 - 13 = 0 \] This simplifies to: \[ 64x^2 - 4 = 0 \]
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Recognize that this can be written as a difference of squares: \[ (8x)^2 - 2^2 = 0 \] This factors to: \[ (8x - 2)(8x + 2) = 0 \]
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Setting each factor equal to zero gives us: \[ 8x - 2 = 0 \quad \text{or} \quad 8x + 2 = 0 \]
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Solving these equations:
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For \( 8x - 2 = 0 \): \[ 8x = 2 \implies x = \frac{2}{8} = \frac{1}{4} \]
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For \( 8x + 2 = 0 \): \[ 8x = -2 \implies x = \frac{-2}{8} = -\frac{1}{4} \]
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The solutions are \( x = \frac{1}{4} \) and \( x = -\frac{1}{4} \).
From the options provided, the correct solutions are:
- \( x = -\frac{1}{4} \)
- \( x = \frac{1}{4} \)
So the correct response is: x equals negative Start Fraction 1 over 4 End Fraction , x equals Start Fraction 1 over 4 End Fraction.