## 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:

**Identify the squares:**4x² is the square of 2x, and 9 is the square of 3.**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.