Algebra, with its intricate equations and complex formulas, can often seem daunting to students. However, understanding the fundamental concepts of algebra is crucial for success in mathematics and various other fields. One such concept is the expansion of (a + b + c)², which holds immense power in simplifying expressions and solving equations. In this article, we will explore the significance of (a + b + c)², its applications in real-world scenarios, and provide step-by-step examples to help you grasp this concept with ease.

## The Basics: Understanding (a + b + c)²

Before delving into the applications of (a + b + c)², let’s first understand what it represents. (a + b + c)² is an algebraic expression that denotes the square of the sum of three variables: a, b, and c. Mathematically, it can be expanded as:

(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

This expansion is derived using the distributive property of multiplication over addition. By multiplying each term within the parentheses by itself and then summing them up, we obtain the expanded form of (a + b + c)².

## Applications of (a + b + c)²

The expansion of (a + b + c)² finds its applications in various fields, including mathematics, physics, and computer science. Let’s explore some of these applications:

### 1. Simplifying Expressions

The expansion of (a + b + c)² allows us to simplify complex algebraic expressions. By substituting the variables with specific values, we can easily evaluate the expression. For example, consider the expression (2x + 3y + 4z)². By expanding it using the formula, we get:

(2x + 3y + 4z)² = (2x)² + (3y)² + (4z)² + 2(2x)(3y) + 2(2x)(4z) + 2(3y)(4z)

This expansion simplifies the expression and enables us to perform further calculations or solve equations more efficiently.

### 2. Solving Equations

The expansion of (a + b + c)² is particularly useful in solving quadratic equations. By rearranging the equation and applying the expansion, we can simplify it and find the roots. For instance, consider the equation x² + 5x + 6 = 0. By rewriting it as (x + 2)(x + 3) = 0 and expanding (x + 2)², we can easily determine the values of x that satisfy the equation.

### 3. Probability Calculations

In probability theory, (a + b + c)² is employed to calculate the probabilities of different outcomes. For example, consider a scenario where three dice are rolled simultaneously. To determine the probability of obtaining a sum of 10, we can expand (a + b + c)², where a, b, and c represent the outcomes of each dice roll. By counting the favorable outcomes and dividing them by the total possible outcomes, we can calculate the probability accurately.

### 4. Geometric Interpretation

The expansion of (a + b + c)² also has a geometric interpretation. It represents the sum of the squares of the three sides of a triangle, along with the sum of the products of the pairs of sides multiplied by two. This interpretation finds applications in geometry, where it helps in calculating areas, perimeters, and other properties of triangles.

## Step-by-Step Examples

Let’s now work through a few step-by-step examples to solidify our understanding of (a + b + c)²:

### Example 1:

Expand (2x + 3y + 4z)².

Solution:

(2x + 3y + 4z)² = (2x)² + (3y)² + (4z)² + 2(2x)(3y) + 2(2x)(4z) + 2(3y)(4z)

= 4x² + 9y² + 16z² + 12xy + 16xz + 24yz

### Example 2:

Solve the equation x² + 6x + 9 = 0.

Solution:

Rewrite the equation as (x + 3)² = 0.

By applying the square root property, we get:

x + 3 = 0

x = -3

Therefore, the solution to the equation is x = -3.

## Q&A

### Q1: Can (a + b + c)² be expanded for more than three variables?

A1: Yes, the expansion of (a + b + c)² can be generalized for any number of variables. The formula for expanding (a₁ + a₂ + a₃ + … + aₙ)² is:

(a₁ + a₂ + a₃ + … + aₙ)² = a₁² + a₂² + a₃² + … + aₙ² + 2(a₁a₂ + a₁a₃ + … + aₙ₋₁aₙ)

### Q2: How is (a + b + c)² related to the binomial theorem?

A2: The expansion of (a + b + c)² is a specific case of the binomial theorem, which provides a formula for expanding the power of a binomial. The binomial theorem states that:

(a + b)ⁿ = C(n, 0)aⁿb⁰ + C(n, 1)aⁿ⁻¹b¹ + C(n, 2)aⁿ⁻²b² + … + C(n, n-1)abⁿ⁻¹ + C(n, n)a⁰bⁿ

where C(n, r) represents the binomial coefficient, given by n! / (r!(n-r)!).

### Q3: Are there any real-world applications of (a + b + c)²?

A3: Yes, the expansion of (a + b + c)² finds applications in various real-world scenarios. For