Identify the factorization of $36-25 x^2$
A. $(5+9 x)(5-9 x)$
B. $(9-5 x)(9+5 x)$
C. $(6+5 x)(6-5 x)$
D. $(5+6 x)(5-6 x)$
Answer (1)
The expression 36 − 25 x 2 is a difference of squares. We recognize that 36 = 6 2 and 25 x 2 = ( 5 x ) 2 . Applying the difference of squares factorization a 2 − b 2 = ( a + b ) ( a − b ) , we get ( 6 + 5 x ) ( 6 − 5 x ) . Therefore, the correct factorization is ( 6 + 5 x ) ( 6 − 5 x ) .
Explanation
Recognizing the Difference of Squares We are asked to identify the correct factorization of the expression 36 − 25 x 2 . This expression is a difference of two squares, which can be factored using the formula a 2 − b 2 = ( a + b ) ( a − b ) .
Identifying a and b We can rewrite the given expression as 6 2 − ( 5 x ) 2 . Here, a = 6 and b = 5 x .
Applying the Factorization Applying the difference of squares factorization, we get ( 6 + 5 x ) ( 6 − 5 x ) .
Examples
The difference of squares factorization is useful in many areas, such as simplifying algebraic expressions, solving equations, and even in engineering. For example, when designing structures, engineers often use the difference of squares to analyze stress distributions and ensure stability. Factoring expressions like 36 − 25 x 2 allows them to simplify complex calculations and make accurate predictions about the behavior of the structure under different loads. This helps in creating safer and more efficient designs.
