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

In algebra, a binomial is an expression with two terms, often involving variables. The square of a binomial, represented as (a + b)², is a common algebraic expression that arises in various mathematical applications.

### The Expansion of (a + b)²

The expression (a + b)² represents the product of (a + b) with itself:

**(a + b)² = (a + b)(a + b)**

To expand this expression, we use the distributive property of multiplication:

**Multiply the first term of the first binomial (a) with each term in the second binomial:**- a * a = a²
- a * b = ab

**Multiply the second term of the first binomial (b) with each term in the second binomial:**- b * a = ab
- b * b = b²

Adding all the terms together, we get:

**(a + b)² = a² + ab + ab + b²**

Combining like terms, we obtain the final expansion:

**(a + b)² = a² + 2ab + b²**

### Understanding the Formula

This formula reveals a pattern:

**The first term:**a² is the square of the first term (a) of the binomial.**The second term:**2ab is twice the product of the first term (a) and the second term (b) of the binomial.**The third term:**b² is the square of the second term (b) of the binomial.

### Applications of the Formula

The formula (a + b)² = a² + 2ab + b² has numerous applications in algebra, geometry, and other areas of mathematics, including:

**Simplifying algebraic expressions:**The formula can be used to simplify expressions involving the square of a binomial.**Factoring expressions:**The formula can be used to factor quadratic expressions.**Solving equations:**The formula can be used to solve equations involving the square of a binomial.**Deriving other formulas:**The formula can be used to derive other algebraic identities, such as the difference of squares formula.

### Example

Let's consider an example:

**(x + 3)²**

Applying the formula:

**(x + 3)² = x² + 2(x)(3) + 3²**

Simplifying:

**(x + 3)² = x² + 6x + 9**

Therefore, the expansion of (x + 3)² is x² + 6x + 9.

### Conclusion

The formula (a + b)² = a² + b² + 2ab is a fundamental algebraic identity that is widely used in various mathematical contexts. Understanding the expansion and application of this formula is essential for mastering algebraic concepts and solving problems involving binomials.