(a-b)(a+b)=a^2-b^2

Jasonbradley

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Jun 16, 2024

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The Difference of Squares: A Powerful Identity

The equation (a - b)(a + b) = a² - b² is a fundamental identity in algebra known as the difference of squares. This identity provides a simple and efficient way to factor or expand expressions involving the difference of two squared terms.

Understanding the Identity

The difference of squares identity arises from the distributive property of multiplication. When we expand the product (a - b)(a + b), we get:

(a - b)(a + b) = a(a + b) - b(a + b)

Applying the distributive property further, we have:

a(a + b) - b(a + b) = a² + ab - ba - b²

Since ab - ba = 0, the expression simplifies to:

a² + ab - ba - b² = a² - b²

Therefore, (a - b)(a + b) = a² - b².

Applications and Examples

The difference of squares identity has numerous applications in various areas of mathematics, including:

• Factoring expressions: We can use this identity to factor expressions containing the difference of two squares. For instance, we can factor x² - 9 as (x - 3)(x + 3).
• Simplifying expressions: The identity can be used to simplify complex expressions by replacing the difference of squares with a simpler form. For example, we can simplify (x² - 4)/(x - 2) as (x - 2)(x + 2)/(x - 2) = x + 2.
• Solving equations: The difference of squares identity can help solve equations where the difference of two squares appears. For example, solving x² - 16 = 0 involves factoring the left side as (x - 4)(x + 4) = 0, leading to the solutions x = 4 and x = -4.

Example: Factoring a Quadratic Expression

Let's factor the quadratic expression 4x² - 9 using the difference of squares identity:

1. Identify the squares: 4x² is the square of 2x, and 9 is the square of 3.
2. Apply the identity: We can write 4x² - 9 as (2x - 3)(2x + 3).

Therefore, the factored form of 4x² - 9 is (2x - 3)(2x + 3).

Conclusion

The difference of squares identity is a valuable tool for simplifying and factoring expressions, and it plays a significant role in various mathematical concepts and applications. Understanding this identity allows us to solve problems efficiently and manipulate algebraic expressions with greater ease.