Gustavo was asked to solve 64x^2+9=13

Question

by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses

A) x=−1/4, x=1/4
x equals negative Start Fraction 1 over 4 End Fraction , x equals Start Fraction 1 over 4 End Fraction

B) x=−1/4
x equals negative Start Fraction 1 over 4 End Fraction

C) x=−3/8, x=3/8
x equals negative Start Fraction 3 over 8 End Fraction , x equals Start Fraction 3 over 8 End Fraction

D) x=3/8

Answer 1

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

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

2. Recognizing this as a difference of squares, we can rewrite it: \[ 64x^2 - 4 = (8x)^2 - (2)^2 \]

3. Applying the difference of squares factoring: \[ (8x - 2)(8x + 2) = 0 \]

4. Setting each factor to zero gives us two equations to solve: \[ 8x - 2 = 0 \quad \text{and} \quad 8x + 2 = 0 \]

5. Solving each equation:

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

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

6. Thus, the solutions are \(x = -\frac{1}{4}\) and \(x = \frac{1}{4}\).

Looking at the responses:

A) \(x = -\frac{1}{4}, x = \frac{1}{4}\) is correct.

B) \(x = -\frac{1}{4}\) is only one part of the solution.

C) \(x = -\frac{3}{8}, x = \frac{3}{8}\) is incorrect.

D) \(x = \frac{3}{8}\) is also incorrect.

The correct answer is:

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