Factor completely 9×2 − 25.
The Correct Answer and Explanation is :
To factor the expression (9x^2 – 25), we first recognize that this is a difference of squares. A difference of squares can be expressed in the form (a^2 – b^2), which factors as ((a – b)(a + b)).
In the expression (9x^2 – 25):
- The term (9x^2) can be rewritten as ((3x)^2).
- The term (25) can be rewritten as (5^2).
Thus, we can identify (a) and (b) as follows:
- (a = 3x)
- (b = 5)
Substituting these values into the difference of squares formula, we get:
[
9x^2 – 25 = (3x)^2 – 5^2 = (3x – 5)(3x + 5)
]
This expression is now fully factored.
To understand why this method works, let’s break it down:
- Identifying Squares: The key to factoring a difference of squares is recognizing the two squared terms. In our case, (9x^2) is a perfect square because it can be expressed as ((3x)^2), and (25) is a perfect square because it equals (5^2).
- Applying the Formula: Once we identify the squares, we can apply the difference of squares formula directly. This is a standard factoring technique in algebra and is very useful for simplifying expressions and solving equations.
- Verification: To ensure our factorization is correct, we can use the distributive property (also known as the FOIL method for binomials) to expand ((3x – 5)(3x + 5)):
[
(3x – 5)(3x + 5) = 3x \cdot 3x + 3x \cdot 5 – 5 \cdot 3x – 5 \cdot 5 = 9x^2 – 25
]
This matches our original expression, confirming that the factorization is indeed correct.
In conclusion, the complete factorization of (9x^2 – 25) is ((3x – 5)(3x + 5)).