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

x=−14
, x=14
x equals negative Start Fraction 1 over 4 End Fraction , x equals Start Fraction 1 over 4 End Fraction
Remaining Attempts : 3

Answer 1

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

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

2. Now, factor the left side. The equation can be rewritten as: \[ 64x^2 - 4 = (8x)^2 - 2^2 \] This is a difference of squares, which can be factored as: \[ (8x - 2)(8x + 2) = 0 \]

3. Now, set each factor to zero: \[ 8x - 2 = 0 \quad \text{or} \quad 8x + 2 = 0 \]

4. Solve each equation for \(x\):

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

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

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