(a-b)^2=a^2-2ab+b^2 Example

4 min read Jun 16, 2024

Understanding the (a-b)^2 Formula: A Step-by-Step Breakdown

The formula (a-b)^2 = a^2 - 2ab + b^2 is a fundamental concept in algebra, and its understanding is crucial for solving various mathematical problems. This article will provide a detailed explanation of this formula, exploring its derivation and practical applications.

What does (a-b)^2 mean?

(a-b)^2 represents the square of the difference between two variables, 'a' and 'b'. In other words, it means multiplying (a-b) by itself.

Deriving the Formula

We can derive the formula by expanding (a-b)^2 using the distributive property:

(a-b)^2 = (a-b)(a-b)

Applying the distributive property:

(a-b)(a-b) = a(a-b) - b(a-b)

Expanding further:

a(a-b) - b(a-b) = a^2 - ab - ba + b^2

Since multiplication is commutative (ab = ba), we can simplify:

a^2 - ab - ba + b^2 = a^2 - 2ab + b^2

Therefore, (a-b)^2 = a^2 - 2ab + b^2

Practical Applications of the Formula

This formula finds numerous applications in various mathematical fields, including:

Algebraic Simplification: The formula allows us to simplify expressions containing squares of differences.
Solving Equations: It can be used to solve quadratic equations and other algebraic equations involving squared differences.
Geometry: It plays a crucial role in proving geometric theorems and finding the area of geometric shapes.
Calculus: The formula is used in differentiating and integrating functions involving squares of differences.

Example

Let's demonstrate the application of the formula with an example:

Problem: Simplify the expression (x-3)^2

Solution: Using the formula, we get:

(x-3)^2 = x^2 - 2(x)(3) + 3^2

Simplifying further:

x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9

Therefore, (x-3)^2 simplifies to x^2 - 6x + 9.

Conclusion

The formula (a-b)^2 = a^2 - 2ab + b^2 is a powerful tool in algebra and various other mathematical fields. Understanding its derivation and applications provides a strong foundation for solving complex mathematical problems. By practicing its application through various examples, you can enhance your proficiency in simplifying algebraic expressions and solving equations.