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 equals Start Fraction 3 over 8 End Fraction

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

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

x=−14
, x=14

Answer 1

To solve the equation \(64x^2 + 9 = 13\) by factoring using the difference of squares pattern, we first need to rearrange the equation.

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

2. Next, we can express this as a difference of squares: \[ (8x)^2 - 2^2 = 0 \]

3. Now, we can factor this using the difference of squares pattern, which states \(a^2 - b^2 = (a - b)(a + b)\): \[ (8x - 2)(8x + 2) = 0 \]

4. Setting each factor to zero gives us the solutions: \[ 8x - 2 = 0 \quad \Rightarrow \quad 8x = 2 \quad \Rightarrow \quad x = \frac{2}{8} = \frac{1}{4} \] and \[ 8x + 2 = 0 \quad \Rightarrow \quad 8x = -2 \quad \Rightarrow \quad x = -\frac{2}{8} = -\frac{1}{4} \]

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

The correct response is: \(x = -\frac{1}{4}, , x = \frac{1}{4}\)