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:
-
\( 4x + 1 = 0 \)
Solving for \( x \):
\[ 4x = -1 \quad \Rightarrow \quad x = -\frac{1}{4} \]
-
\( 4x - 1 = 0 \)
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:
\( x = -\frac{1}{4}, x = \frac{1}{4} \).