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

Expanding and simplifying algebraic expressions using the formula of (a – b)² allows us to manipulate and solve equations more efficiently.

### 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 see how:

Area of the larger square = a²

Area of the smaller square = b²

Difference in areas = a² – b²

By using the formula of (a – b)², we can easily determine the difference in areas between two squares or any other geometric figures.

### Calculus

The formula of (a – b)² is also relevant in calculus, particularly in the context of differentiation. When differentiating a function that involves a square of a difference, we can apply the formula to simplify the expression before proceeding with the differentiation process. This simplification helps in reducing complexity and making the differentiation process more manageable.

## Examples of (a – b)² in Action

Let’s explore a few examples to understand how the formula of (a – b)² can be applied in different scenarios:

### Example 1: Algebraic Simplification

Consider the expression (2x – 3y)². To simplify this expression, we can use the formula of (a – b)²:

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

= 4x² – 12xy + 9y²

By applying the formula, we have simplified the expression (2x – 3y)² to 4x² – 12xy + 9y².

### Example 2: Geometry

Consider two squares, one with side length 5 units and another with side length 3 units. To find the difference in their areas, we can use the formula of (a – b)²:

Area of the larger square = (5 units)² = 25 square units

Area of the smaller square = (3 units)² = 9 square units

Difference in areas = 25 square units – 9 square units = 16 square units

Therefore, the difference in areas between the two squares is 16 square units.

### Example 3: Calculus

Consider the function f(x) = (2x – 1)². To differentiate this function, we can simplify it using the formula of (a – b)²:

f(x) = (2x – 1)² = (2x)² – 2(2x)(1) + (1)²

= 4x² – 4x + 1

Now, we can differentiate the simplified function f(x) = 4x² – 4x + 1 using the rules of calculus.

## Key Takeaways

- The formula of (a – b)² is used to find the square of the difference between two numbers, a and b.
- The formula can be derived using the distributive property of multiplication over addition.
- Applications of the formula include algebraic simplification, geometry, and calculus.
- By applying the formula, we can simplify algebraic expressions, find the difference in areas of geometric figures, and simplify functions before differentiation.