## Understanding the (a + b + c)^2 Formula

The formula (a + b + c)^2 is a fundamental concept in algebra and finds its applications in various mathematical and real-world problems. It allows us to quickly expand expressions involving the sum of three terms squared. Let's dive deeper into this formula and understand its usage with examples.

### The Formula:

The formula states that:

**(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc**

This means that squaring the sum of three terms results in the sum of the squares of each term plus twice the product of each pair of terms.

### Example 1:

**Problem:** Expand (x + y + 2)^2

**Solution:**

Using the formula, we get:

(x + y + 2)^2 = x^2 + y^2 + 2^2 + 2(x)(y) + 2(x)(2) + 2(y)(2)

Simplifying further:

**(x + y + 2)^2 = x^2 + y^2 + 4 + 2xy + 4x + 4y**

### Example 2:

**Problem:** Calculate the value of (3 + 4 + 5)^2

**Solution:**

We can directly apply the formula:

(3 + 4 + 5)^2 = 3^2 + 4^2 + 5^2 + 2(3)(4) + 2(3)(5) + 2(4)(5)

Simplifying:

(3 + 4 + 5)^2 = 9 + 16 + 25 + 24 + 30 + 40

**(3 + 4 + 5)^2 = 144**

### Why is this Formula Useful?

The (a + b + c)^2 formula simplifies the process of expanding complex expressions. It avoids the tedious step of multiplying the entire expression by itself, saving time and effort. Additionally, the formula plays a crucial role in various mathematical applications, including:

**Factoring quadratic expressions:**Understanding this formula helps in factoring expressions containing three terms squared.**Solving equations:**It can be used to simplify equations involving the square of a sum of three terms.**Geometry:**The formula finds its application in deriving formulas related to areas and volumes of geometric shapes.

Understanding the (a + b + c)^2 formula is essential for anyone working with algebraic expressions and can be a valuable tool in various mathematical and scientific applications.