## The Difference of Squares Formula: (x-y)(x+y)

The formula **(x-y)(x+y) = x² - y²** is a fundamental algebraic identity known as the **difference of squares formula**. This formula provides a shortcut for expanding and simplifying expressions involving the difference of two squares.

### Understanding the Formula

The difference of squares formula states that the product of the sum and difference of two terms is equal to the difference of their squares. Let's break down the components:

**(x-y):**This represents the difference of two terms, 'x' and 'y'.**(x+y):**This represents the sum of the same two terms, 'x' and 'y'.**x² - y²:**This represents the difference of the squares of the two terms, 'x' and 'y'.

### How the Formula Works

We can understand the formula by expanding the left-hand side using the distributive property (also known as FOIL):

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

Notice that the middle terms, '+xy' and '-xy', cancel out, leaving us with the difference of squares.

### Applications of the Difference of Squares Formula

The difference of squares formula has numerous applications in algebra and beyond. Some common uses include:

**Factoring expressions:**The formula helps us factor expressions that resemble the difference of squares. For instance, we can factor x² - 9 as (x-3)(x+3).**Simplifying equations:**It can be used to simplify equations by replacing the difference of squares with a single term. For example, we can simplify the equation x² - 4 = 0 by rewriting it as (x-2)(x+2) = 0.**Solving problems in geometry:**The formula can be used to find areas, volumes, and other geometric properties involving squares.**Higher level mathematics:**The difference of squares formula is a fundamental concept that appears in various areas of higher mathematics, including calculus and abstract algebra.

### Example

**Factor the expression x⁴ - 16**

**Solution:**

We can recognize x⁴ - 16 as the difference of squares, where x² is the first term and 4² is the second term. Applying the formula, we get:

x⁴ - 16 = (x²)² - 4² = (x² - 4)(x² + 4)

Now, we can further factor (x² - 4) as the difference of squares:

(x² - 4)(x² + 4) = (x-2)(x+2)(x² + 4)

Therefore, the completely factored form of x⁴ - 16 is **(x-2)(x+2)(x² + 4)**.

### Conclusion

The difference of squares formula is a powerful tool in algebra, providing a shortcut for simplifying and factoring expressions. It's essential for understanding various algebraic concepts and solving problems across different mathematical fields. By recognizing this formula and its applications, you can enhance your algebraic skills and approach problem-solving more effectively.