To determine which of the given expressions can be factored using the difference of squares method, let’s review the difference of squares formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
We need to identify the expression that fits this pattern. Let’s examine each option:
Option A: [tex]\( 9x - 16y \)[/tex]
This expression has two terms: [tex]\( 9x \)[/tex] and [tex]\( 16y \)[/tex]. However, neither term is a perfect square, and it does not fit the form [tex]\( a^2 - b^2 \)[/tex]. Therefore, it cannot be factored using the difference of squares method.
Option B: [tex]\( 9x^2 - 16y^2 \)[/tex]
This is a two-term expression. Let’s check if each term is a perfect square:
- [tex]\( 9x^2 = (3x)^2 \)[/tex]
- [tex]\( 16y^2 = (4y)^2 \)[/tex]
Both terms are perfect squares. This fits the form [tex]\( a^2 - b^2 \)[/tex] where [tex]\( a = 3x \)[/tex] and [tex]\( b = 4y \)[/tex]. We can factor it as:
[tex]\[
9x^2 - 16y^2 = (3x)^2 - (4y)^2 = (3x - 4y)(3x + 4y)
\][/tex]
So, option B can be factored using the difference of squares method.
Option C: [tex]\( 9x^2 + 16y^2 \)[/tex]
This expression has two terms: [tex]\( 9x^2 \)[/tex] and [tex]\( 16y^2 \)[/tex], both of which are perfect squares. However, it involves a sum of squares, not a difference. The difference of squares method does not apply to sums of squares. Therefore, it cannot be factored using the difference of squares method.
Option D: [tex]\( 9x + 16y \)[/tex]
This expression has two terms: [tex]\( 9x \)[/tex] and [tex]\( 16y \)[/tex]. Neither term is a perfect square, and it does not fit the form [tex]\( a^2 - b^2 \)[/tex]. Therefore, it cannot be factored using the difference of squares method.
Based on the analysis, only option B can be factored using the difference of squares method. The answer is:
[tex]\[
\boxed{B}
\][/tex]
