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)

Question

Responses

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
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

The equation given is \( 64x^2 + 9 = 13 \).

First, we can rewrite it by moving 13 to the left side:

\[ 64x^2 + 9 - 13 = 0 \]

This simplifies to:

\[ 64x^2 - 4 = 0 \]

Now we can factor this expression. Notice that \( 64x^2 - 4 \) can be recognized as a difference of squares:

\[ (8x)^2 - 2^2 = 0 \]

Using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \), we get:

\[ (8x - 2)(8x + 2) = 0 \]

Setting each factor equal to zero gives us the equations to solve for \( x \):

1. \( 8x - 2 = 0 \)

\[ 8x = 2 \implies x = \frac{2}{8} = \frac{1}{4} \]

2. \( 8x + 2 = 0 \)

\[ 8x = -2 \implies x = -\frac{2}{8} = -\frac{1}{4} \]

Thus, the solutions are:

\[ x = \frac{1}{4}, \quad x = -\frac{1}{4} \]

The correct response from the options provided would be:

\( x = -\frac{1}{4}, \quad x = \frac{1}{4} \)