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

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

x=−14

1 answer

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

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

  2. Now, simplify: \[ 64x^2 - 4 = 0 \]

  3. Notice that \( 64x^2 - 4 \) can be factored using the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \) where \( a = 8x \) and \( b = 2 \): \[ 64x^2 - 4 = (8x)^2 - (2)^2 = (8x - 2)(8x + 2) = 0 \]

  4. Set each factor equal to zero: \[ 8x - 2 = 0 \quad \text{or} \quad 8x + 2 = 0 \]

  5. Solving these equations:

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

Thus, the solutions are:

  • \( x = \frac{1}{4} \)
  • \( x = -\frac{1}{4} \)

Among the given options, the closest correct response is:

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