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.

Question

GPT-4o mini · 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} $**.