Gustavo was asked to solve 64x2+9=13

Question

by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses

x=−14
, x=14
x equals negative Start Fraction 1 over 4 End Fraction , x equals Start Fraction 1 over 4 End Fraction

x=−14
x equals negative Start Fraction 1 over 4 End Fraction

x=38
x equals Start Fraction 3 over 8 End Fraction

x=−38
, x=38

Answer 1

To solve the equation \( 64x^2 + 9 = 13 \) using factoring and the difference of squares pattern, we first need to rewrite the equation in standard form:

1. Move \( 13 \) to the left side: \[ 64x^2 + 9 - 13 = 0 \] This simplifies to: \[ 64x^2 - 4 = 0 \]

2. Now, factor the left side using the difference of squares pattern: \[ (8x)^2 - (2)^2 = 0 \] This can be factored as: \[ (8x - 2)(8x + 2) = 0 \]

3. Set each factor equal to zero: \[ 8x - 2 = 0 \quad \text{and} \quad 8x + 2 = 0 \]

4. Solve each equation:

   • For \( 8x - 2 = 0 \): \[ 8x = 2 \implies x = \frac{2}{8} = \frac{1}{4} \]

   • For \( 8x + 2 = 0 \): \[ 8x = -2 \implies x = \frac{-2}{8} = -\frac{1}{4} \]

So the solutions are: \[ x = \frac{1}{4} \quad \text{and} \quad x = -\frac{1}{4} \]

Therefore, the correct solution(s) from your responses are:

• \( x = -\frac{1}{4} \)
• \( x = \frac{1}{4} \)