Mathematics is a fascinating subject that encompasses a wide range of concepts and formulas. One such formula that holds immense significance is the formula of (a – b)², also known as the formula for the square of a difference. This formula plays a crucial role in various mathematical applications, including algebra, geometry, and calculus. In this article, we will delve into the intricacies of this formula, explore its applications, and provide valuable insights to help you understand and apply it effectively.

## What is the Formula of (a – b)²?

The formula of (a – b)² is a mathematical expression used to find the square of the difference between two numbers, a and b. It can be represented as:

(a – b)² = a² – 2ab + b²

This formula can be derived using the distributive property of multiplication over addition. By expanding (a – b)², we get:

(a – b)² = (a – b)(a – b)

= a(a – b) – b(a – b)

= a² – ab – ba + b²

= a² – 2ab + b²

It is important to note that the formula of (a – b)² is a special case of the more general formula for the square of a binomial, (a + b)². The only difference between the two formulas is the sign of the middle term. In (a – b)², the middle term is negative, whereas in (a + b)², the middle term is positive.

## Applications of the Formula of (a – b)²

The formula of (a – b)² finds extensive applications in various branches of mathematics. Let’s explore some of its key applications:

### Algebraic Simplification

The formula of (a – b)² is often used to simplify algebraic expressions. By applying this formula, we can expand and simplify expressions involving squares of differences. For example, consider the expression (3x – 2y)². Using the formula, we can expand it as:

(3x – 2y)² = (3x)² – 2(3x)(2y) + (2y)²

= 9x² – 12xy + 4y²

This expansion allows us to simplify the expression and perform further algebraic manipulations if required.

### Geometry

The formula of (a – b)² is also applicable in geometry, particularly in the context of finding the difference of areas. Consider a square with side length ‘a’ and another square with side length ‘b’. The difference in their areas can be calculated using the formula of (a – b)². Let’s illustrate this with an example:

Suppose we have a square with side length 5 units and another square with side length 3 units. The difference in their areas can be calculated as:

(5 – 3)² = 5² – 2(5)(3) + 3²

= 25 – 30 + 9

= 4

Therefore, the difference in the areas of the two squares is 4 square units.

### Calculus

In calculus, the formula of (a – b)² is used in various applications, such as finding the derivative of functions involving squares of differences. For instance, consider the function f(x) = (x – 2)². To find its derivative, we can apply the formula of (a – b)² and differentiate each term:

f(x) = (x – 2)²

= x² – 2(2)(x) + 2²

= x² – 4x + 4

Now, we can differentiate the function f(x) = x² – 4x + 4 with respect to x:

f'(x) = 2x – 4

This derivative provides valuable information about the rate of change of the function at any given point.

## Examples and Case Studies

To further illustrate the practical applications of the formula of (a – b)², let’s consider a few examples and case studies:

### Example 1: Profit Calculation

Suppose a company’s profit in the first quarter of the year was $10,000, and its profit in the second quarter was $8,000. To calculate the difference in profits, we can use the formula of (a – b)²:

(10,000 – 8,000)² = 10,000² – 2(10,000)(8,000) + 8,000²

= 100,000,000 – 160,000,000 + 64,000,000

= 4,000,000

Therefore, the difference in profits between the two quarters is $4,000,000.

### Case Study: Population Growth

Let’s consider a case study involving population growth. Suppose the population of a city in 2010 was 500,000, and it increased to 700,000 in 2020. To calculate the difference in population, we can apply the formula of (a – b)²:

(700,000 – 500,000)² = 700,000² – 2(700,000)(500,000) + 500,000²

= 490,000,000,000 – 700,000,000,000 + 250,000,000,000

= 40,000,000,000

Therefore, the difference in population between 2010 and 2020 is 40,000,000.

## Key Takeaways

The formula of (a – b)² is a powerful mathematical tool that finds applications in various fields. Here are the key takeaways from this article:

- The formula of (a – b)² is derived using the distributive property of multiplication over addition.
- It