## The Square of a Trinomial: Understanding (a+b+c)^2

The formula **(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca** is a fundamental algebraic identity that helps us expand the square of a trinomial. Understanding this identity is crucial in various mathematical applications, particularly in algebra and calculus.

### Expanding the Square

The formula can be derived by applying the distributive property of multiplication. Let's break it down:

**Start with (a+b+c)^2:**This represents the square of the trinomial (a+b+c).**Expand the square:**We can rewrite it as (a+b+c) * (a+b+c).**Apply the distributive property:**We multiply each term in the first trinomial by each term in the second trinomial.

This results in the following expansion:

```
(a+b+c) * (a+b+c) = a(a+b+c) + b(a+b+c) + c(a+b+c)
```

**Distribute again:**

```
= a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2
```

**Combine like terms:**

```
= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
```

This demonstrates how the formula **(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca** is derived.

### Applications and Significance

The square of a trinomial formula has various applications, including:

**Simplifying algebraic expressions:**It helps simplify expressions involving squares of trinomials.**Solving equations:**It can be used to solve equations that involve squares of trinomials.**Factoring expressions:**It can be used to factor expressions that involve squares of trinomials.**Geometric applications:**It can be used to solve problems involving areas and volumes of geometric shapes.

### Example

Let's apply the formula to a practical example:

**Problem:** Expand (2x + 3y + 5)^2

**Solution:** Using the formula, we get:

(2x + 3y + 5)^2 = (2x)^2 + (3y)^2 + (5)^2 + 2(2x)(3y) + 2(2x)(5) + 2(3y)(5)

Simplifying the expression:

= 4x^2 + 9y^2 + 25 + 12xy + 20x + 30y

Therefore, (2x + 3y + 5)^2 expands to 4x^2 + 9y^2 + 25 + 12xy + 20x + 30y.

### Conclusion

The square of a trinomial formula is an essential tool in algebra and other mathematical disciplines. Understanding its derivation and applications allows for efficient simplification, problem-solving, and a deeper understanding of mathematical concepts. By applying this formula, we can simplify complex expressions and solve various problems involving trinomials.