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Solve 4x^2−64=0

using the difference of squares pattern.(1 point)
Responses

x=−4
x equals negative 4

x=8
, x=−8
x equals 8 , x equals negative 8

x=4
x equals 4

x=4
, x=−4

1 answer

To solve the equation \( 4x^2 - 64 = 0 \) using the difference of squares pattern, we can first rewrite it in a form that highlights the difference of squares.

  1. Start with the equation: \[ 4x^2 - 64 = 0 \]

  2. Factor out the common term: \[ 4(x^2 - 16) = 0 \]

  3. Now, we can use the difference of squares. Recall that \( a^2 - b^2 = (a - b)(a + b) \). Here, \( x^2 - 16 \) can be expressed as: \[ x^2 - 4^2 \]

  4. Apply the difference of squares: \[ 4(x - 4)(x + 4) = 0 \]

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

  6. Solve for \( x \): \[ x = 4 \quad \text{or} \quad x = -4 \]

Thus, the solutions to the equation \( 4x^2 - 64 = 0 \) are: \[ x = 4, \quad x = -4 \]

The responses matching this solution are:

  • \( x = 4, ; x = -4 \)
  • \( x \text{ equals } 4 ; \text{and} ; x \text{ equals } negative 4 \)

So the correct options would be: \( x = 4, ; x = -4 \)