%5E2-(a-1)%5E2-3(a%2B1)(a-1).jpeg "(a+1)^2-(a-1)^2-3(a+1)(a-1)")
Simplifying the Expression (a+1)²-(a-1)²-3(a+1)(a-1)
This article will explore the simplification of the algebraic expression (a+1)²-(a-1)²-3(a+1)(a-1). We will utilize algebraic identities and basic arithmetic operations to achieve a simplified form.
Understanding the Expression
The expression involves three terms:
- (a+1)²: This is a perfect square trinomial, expanding to a² + 2a + 1
- (a-1)²: This is also a perfect square trinomial, expanding to a² - 2a + 1
- 3(a+1)(a-1): This represents the product of three factors, where (a+1) and (a-1) form the difference of squares pattern.
Simplifying Using Identities
Let's utilize the following algebraic identities to simplify the expression:
- Difference of Squares: (x+y)(x-y) = x² - y²
- Perfect Square Trinomial: (x+y)² = x² + 2xy + y²
- Perfect Square Trinomial: (x-y)² = x² - 2xy + y²
Step 1: Expanding the squares
- (a+1)² = a² + 2a + 1
- (a-1)² = a² - 2a + 1
Step 2: Applying the Difference of Squares Identity
- 3(a+1)(a-1) = 3(a² - 1²) = 3a² - 3
Step 3: Substituting the expanded terms into the original expression
- (a² + 2a + 1) - (a² - 2a + 1) - (3a² - 3)
Step 4: Simplifying by combining like terms
- a² + 2a + 1 - a² + 2a - 1 - 3a² + 3
- -3a² + 4a + 3
Conclusion
Therefore, the simplified form of the expression (a+1)²-(a-1)²-3(a+1)(a-1) is -3a² + 4a + 3. This process demonstrates how understanding algebraic identities can significantly simplify complex expressions.