Question

Gustavo was asked to solve 64x2+9=13
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=−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

Answer 1

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

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

2. Recognize that this can be written as a difference of squares:
\[
(8x)^2 - 2^2 = 0
\]
This factors to:
\[
(8x - 2)(8x + 2) = 0
\]

3. Setting each factor equal to zero gives us:
\[
8x - 2 = 0 \quad \text{or} \quad 8x + 2 = 0
\]

4. Solving these equations:
- For \( 8x - 2 = 0 \):
\[
8x = 2 \implies x = \frac{2}{8} = \frac{1}{4}
\]

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

5. 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**.