%5E2-(5%2F2a-7%2F2b)%5E2.jpeg "(7/2a-5/2b)^2-(5/2a-7/2b)^2")
Simplifying the Expression: (7/2a - 5/2b)^2 - (5/2a - 7/2b)^2
This article will guide you through simplifying the expression (7/2a - 5/2b)^2 - (5/2a - 7/2b)^2. We'll use algebraic manipulation and some key identities to achieve a concise result.
Recognizing the Pattern
The given expression looks a bit intimidating at first, but there's a pattern we can exploit. Notice that both terms are squared differences. This suggests we can utilize the "difference of squares" factorization:
(a - b)^2 - (c - d)^2 = (a - b + c - d)(a - b - c + d)
Applying the Factorization
Let's apply this factorization to our problem:
- a = 7/2a
- b = 5/2b
- c = 5/2a
- d = 7/2b
Substituting these values into the factorization, we get:
(7/2a - 5/2b + 5/2a - 7/2b)(7/2a - 5/2b - 5/2a + 7/2b)
Simplifying Further
Now we can combine like terms:
(12/2a - 12/2b)(2/2b - 2/2a)
Simplifying the fractions:
(6a - 6b)(b - a)
Final Result
Therefore, the simplified expression is (6a - 6b)(b - a). This result is much more manageable than the original expression, and it highlights the importance of recognizing patterns in algebraic expressions.