Ginger was asked to solve 16x2−1 by using the difference of two squares pattern. What is/are the correct solution(s)?(1 point) Responses x=14 x is equal to 1 fourth x=116 x is equal to 1 over 16 x=−14 , x=14 x is equal to negative 1 fourth, x is equal to 1 fourth There is no solution for x because the difference of squares pattern does not apply. There is no solution for x because the difference of squares pattern does not apply.

1 answer

To solve the expression \( 16x^2 - 1 \) using the difference of squares pattern, we can recognize that it fits the form \( a^2 - b^2 \), where:

  • \( a^2 = (4x)^2 \)
  • \( b^2 = 1^2 \)

Thus, we can rewrite \( 16x^2 - 1 \) as:

\[ (4x)^2 - 1^2 \]

According to the difference of squares formula, \( a^2 - b^2 = (a+b)(a-b) \), we can express this as:

\[ (4x + 1)(4x - 1) \]

To find the solutions for \( x \), we can set each factor equal to zero:

  1. \( 4x + 1 = 0 \)

    Solving for \( x \):

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

  2. \( 4x - 1 = 0 \)

    Solving for \( x \):

    \[ 4x = 1 \quad \Rightarrow \quad x = \frac{1}{4} \]

Therefore, the solutions to the equation \( 16x^2 - 1 = 0 \) are:

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

Thus, the correct response among the provided choices is:

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