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=−38
, x=38
x equals negative Start Fraction 3 over 8 End Fraction , x equals Start Fraction 3 over 8 End Fraction

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

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

x=−14
, x=14

Answer 1

To solve the equation \( 64x^2 + 9 = 13 \) by factoring, we first need to rearrange it to a standard form.

1. Subtract 13 from both sides: \[ 64x^2 + 9 - 13 = 0 \] \[ 64x^2 - 4 = 0 \]

2. Now, we can see that this is a difference of squares: \[ (8x)^2 - 2^2 = 0 \]

3. Using the difference of squares factorization formula \( a^2 - b^2 = (a-b)(a+b) \), we can factor: \[ (8x - 2)(8x + 2) = 0 \]

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

5. Solving for \( x \) gives: \[ 8x - 2 = 0 \implies 8x = 2 \implies x = \frac{2}{8} = \frac{1}{4} \] \[ 8x + 2 = 0 \implies 8x = -2 \implies x = \frac{-2}{8} = -\frac{1}{4} \]

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

Considering the provided options, the correct one is:

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