## Understanding the Expansion of (a + b)^2

The formula **(a + b)^2 = a^2 + 2ab + b^2** is a fundamental algebraic identity that arises from the distributive property of multiplication. This formula helps us simplify expressions involving squares of binomials and is a key concept in algebra.

### Derivation of the Formula

The expansion of (a + b)^2 is derived by applying the distributive property twice:

**Expanding the square:**(a + b)^2 = (a + b) * (a + b)**Distributing the first term:**(a + b) * (a + b) = a * (a + b) + b * (a + b)**Distributing again:**a * (a + b) + b * (a + b) = a^2 + ab + ba + b^2**Combining like terms:**a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

Therefore, we arrive at the final result: **(a + b)^2 = a^2 + 2ab + b^2**

### Applications of the Formula

This formula has various applications in algebra, including:

**Simplifying expressions:**The formula can be used to simplify complex algebraic expressions involving squares of binomials.**Solving equations:**The formula can be used to solve equations containing squares of binomials.**Factoring expressions:**The formula can be used to factor expressions involving squares of binomials.

### Example

Let's consider an example to illustrate the application of the formula:

**Simplify the expression (x + 3)^2:**

Using the formula (a + b)^2 = a^2 + 2ab + b^2, we can expand the expression as:

(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 formula (a + b)^2 = a^2 + 2ab + b^2 is a powerful tool in algebra that simplifies expressions, solves equations, and helps with factoring. Understanding this formula is crucial for mastering algebraic concepts and solving various mathematical problems.