Answer :
To solve the expression using the difference of two squares, we start by understanding the formula involved. The difference of two squares formula is:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Now, let's look at the options provided to see which one matches this form.
Option Analysis:
- [tex]\((-7x)(4+7x)\)[/tex]: This expands to [tex]\(-28x - 49x^2\)[/tex], which does not fit the form [tex]\(a^2 - b^2\)[/tex].
- [tex]\(16 - 25x^2\)[/tex]: This can be viewed as [tex]\((4)^2 - (5x)^2\)[/tex], which fits the pattern [tex]\(a^2 - b^2\)[/tex] where [tex]\(a = 4\)[/tex] and [tex]\(b = 5x\)[/tex].
- [tex]\(64 - 25x^2\)[/tex]: This can be rewritten as [tex]\((8)^2 - (5x)^2\)[/tex], fitting the pattern with [tex]\(a = 8\)[/tex] and [tex]\(b = 5x\)[/tex].
- [tex]\(16 - 49x^2\)[/tex]: This matches [tex]\((4)^2 - (7x)^2\)[/tex], which is a difference of squares with [tex]\(a = 4\)[/tex] and [tex]\(b = 7x\)[/tex].
- [tex]\(16 + 56x - 49x^2\)[/tex]: This option expands and does not fit the form of a difference of squares.
- [tex]\(8 - 49x^2\)[/tex]: This can be interpreted as [tex]\((\sqrt{8})^2 - (7x)^2\)[/tex], but it's not an exact integer square for the first term and does not fit neatly into the difference of squares.
Among these, [tex]\(16 - 49x^2\)[/tex] is the expression that cleanly utilizes the difference of squares, following the standard formula [tex]\(a^2 - b^2\)[/tex].
Therefore, the correct simplification using the difference of squares for this expression is:
[tex]\(16 - 49x^2\)[/tex]
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Now, let's look at the options provided to see which one matches this form.
Option Analysis:
- [tex]\((-7x)(4+7x)\)[/tex]: This expands to [tex]\(-28x - 49x^2\)[/tex], which does not fit the form [tex]\(a^2 - b^2\)[/tex].
- [tex]\(16 - 25x^2\)[/tex]: This can be viewed as [tex]\((4)^2 - (5x)^2\)[/tex], which fits the pattern [tex]\(a^2 - b^2\)[/tex] where [tex]\(a = 4\)[/tex] and [tex]\(b = 5x\)[/tex].
- [tex]\(64 - 25x^2\)[/tex]: This can be rewritten as [tex]\((8)^2 - (5x)^2\)[/tex], fitting the pattern with [tex]\(a = 8\)[/tex] and [tex]\(b = 5x\)[/tex].
- [tex]\(16 - 49x^2\)[/tex]: This matches [tex]\((4)^2 - (7x)^2\)[/tex], which is a difference of squares with [tex]\(a = 4\)[/tex] and [tex]\(b = 7x\)[/tex].
- [tex]\(16 + 56x - 49x^2\)[/tex]: This option expands and does not fit the form of a difference of squares.
- [tex]\(8 - 49x^2\)[/tex]: This can be interpreted as [tex]\((\sqrt{8})^2 - (7x)^2\)[/tex], but it's not an exact integer square for the first term and does not fit neatly into the difference of squares.
Among these, [tex]\(16 - 49x^2\)[/tex] is the expression that cleanly utilizes the difference of squares, following the standard formula [tex]\(a^2 - b^2\)[/tex].
Therefore, the correct simplification using the difference of squares for this expression is:
[tex]\(16 - 49x^2\)[/tex]
