## Factoring the Difference of Squares: (x² - y²)

The expression (x² - y²) is a classic example of a **difference of squares**. This pattern appears frequently in algebra and can be factored using a simple and elegant formula.

### Understanding the Pattern

The difference of squares pattern arises when we have two perfect squares being subtracted from each other. In the expression (x² - y²), we have:

**x²:**The square of the variable x**y²:**The square of the variable y

### Factoring the Difference of Squares

The formula for factoring the difference of squares is:

**(a² - b²) = (a + b)(a - b)**

To apply this to (x² - y²), we simply substitute:

**a = x****b = y**

Therefore, the factored form of (x² - y²) is:

**(x² - y²) = (x + y)(x - y)**

### Example

Let's factor the expression (9x² - 4y²):

**Identify the perfect squares:**We have 9x² (which is (3x)²) and 4y² (which is (2y)²)**Apply the formula:**(9x² - 4y²) = (3x + 2y)(3x - 2y)

### Why Does This Work?

The formula for factoring the difference of squares works because of the distributive property. When you expand (x + y)(x - y), you get:

- x(x - y) + y(x - y)
- x² - xy + xy - y²
- x² - y²

The middle terms (-xy + xy) cancel out, leaving us with the original expression (x² - y²).

### Applications

The difference of squares pattern is used extensively in algebra and other areas of mathematics. Some common applications include:

**Solving quadratic equations:**By factoring a quadratic equation into the difference of squares, we can easily find its roots.**Simplifying expressions:**Factoring the difference of squares can simplify complex algebraic expressions.**Solving problems in geometry:**The difference of squares pattern can be used to find the area or volume of certain geometric shapes.

Understanding the difference of squares pattern is a crucial step in developing strong algebraic skills. By mastering this pattern, you'll be better equipped to tackle more complex mathematical problems.