## The "Difference of Squares" Formula: A Key Algebraic Tool

In mathematics, the **difference of squares formula** is a fundamental algebraic identity that simplifies the expansion of the square of the difference of two terms. It states:

**(a - b)² = a² - 2ab + b²**

This formula holds true for any real numbers a and b. It essentially expands the square of a binomial expression, yielding a trinomial with specific coefficients.

### Understanding the Formula

The formula is derived from the distributive property of multiplication. When expanding (a - b)², we multiply the binomial by itself:

(a - b)² = (a - b)(a - b)

Applying the distributive property, we get:

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

Since multiplication is commutative (ab = ba), we can simplify further:

a² - ab - ba + b² = **a² - 2ab + b²**

### Applications and Significance

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

**Factoring expressions:**It allows us to factor quadratic expressions into two binomials, simplifying the process of finding solutions.**Solving equations:**By factoring equations using the difference of squares formula, we can find their roots or solutions more easily.**Simplifying expressions:**This formula helps in simplifying complex algebraic expressions by eliminating squares and simplifying terms.

### Example

Consider the expression (x - 3)². Using the difference of squares formula:

(x - 3)² = x² - 2(x)(3) + 3² = x² - 6x + 9

This formula is essential for understanding and manipulating algebraic expressions, leading to quicker and more efficient solutions in various mathematical contexts.