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

x=38

Answer 1

To solve the equation \(64x^2 + 9 = 13\) by factoring using the difference of squares, let's start by rewriting the equation:

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

2. This can be rewritten as:
\[
64x^2 - 4 = (8x)^2 - 2^2
\]

3. Now we can apply the difference of squares factoring pattern, \(a^2 - b^2 = (a - b)(a + b)\):
\[
(8x - 2)(8x + 2) = 0
\]

4. Set each factor to zero:
\[
8x - 2 = 0 \implies 8x = 2 \implies x = \frac{1}{4}
\]
\[
8x + 2 = 0 \implies 8x = -2 \implies x = -\frac{1}{4}
\]

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

The correct set of answers from the options you provided is:
\[
x = -\frac{1}{4}, \; x = \frac{1}{4}
\]