## Understanding the Square of a Binomial: (a - b)^2 = a^2 - 2ab + b^2

The equation **(a - b)^2 = a^2 - 2ab + b^2** represents a fundamental algebraic identity that defines the square of a binomial. This identity is essential for simplifying expressions, solving equations, and understanding various mathematical concepts.

### What is a Binomial?

A binomial is a mathematical expression consisting of two terms, typically connected by addition or subtraction. In the equation above, **(a - b)** represents a binomial where 'a' and 'b' are variables or constants.

### Expanding the Square

To understand the identity, let's break down the process of squaring a binomial:

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

Now, we can expand this using the distributive property (also known as FOIL - First, Outer, Inner, Last):

**First:**a * a = a^2**Outer:**a * (-b) = -ab**Inner:**(-b) * a = -ab**Last:**(-b) * (-b) = b^2

Combining these terms, we get:

**a^2 - ab - ab + b^2 = a^2 - 2ab + b^2**

This demonstrates how the identity **(a - b)^2 = a^2 - 2ab + b^2** is derived.

### Applications of the Identity

This identity has numerous applications in mathematics, including:

**Simplifying algebraic expressions:**The identity can be used to quickly simplify expressions involving the square of a binomial.**Solving equations:**By applying the identity, we can transform equations into simpler forms, making them easier to solve.**Factoring polynomials:**The identity helps us factor quadratic expressions by recognizing patterns of the form**a^2 - 2ab + b^2**.**Geometry:**The identity can be applied in geometric problems involving squares and rectangles.

### Example:

Let's say we want to simplify the expression (x - 3)^2:

Using the identity:

**(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9**

Therefore, the simplified expression is **x^2 - 6x + 9**.

### Conclusion:

The identity **(a - b)^2 = a^2 - 2ab + b^2** is a fundamental algebraic principle that has numerous applications in simplifying expressions, solving equations, and understanding various mathematical concepts. By understanding and applying this identity, you can efficiently manipulate algebraic expressions and solve a wide range of mathematical problems.