Question

Use the Distributive Property to find which expression is equivalent to 9x²−25.
Responses:
1. (3x+5)²
2. 9x(x−25)
3. (4.5x+12.5)(4.5x−12.5)
4. (3x+5)(3x−5)

Answer 1

To use the Distributive Property to find an equivalent expression for \( 9x^2 - 25 \), we can recognize that it is a difference of squares. This can be factored using the formula \( a^2 - b^2 = (a + b)(a - b) \).

In this case:
- \( 9x^2 \) is \( (3x)^2 \)
- \( 25 \) is \( 5^2 \)

Thus, we can write:

\[
9x^2 - 25 = (3x)^2 - 5^2 = (3x + 5)(3x - 5)
\]

Now, let's analyze the given options:

1. \( (3x + 5)^2 \) - This is not correct as it expands to \( 9x^2 + 30x + 25 \).
2. \( 9x(x - 25) \) - This is not correct because it expands to \( 9x^2 - 225x \).
3. \( (4.5x + 12.5)(4.5x - 12.5) \) - This can be checked, but it does not simplify cleanly to \( 9x^2 - 25 \).
4. \( (3x + 5)(3x - 5) \) - This is the correct factoring of \( 9x^2 - 25 \).

Thus, the expression that is equivalent to \( 9x^2 - 25 \) is:

**4. \( (3x + 5)(3x - 5) \)**.