(a-b)2=a2-2ab+b2 Answer

3 min read Jun 16, 2024
(a-b)2=a2-2ab+b2 Answer

The Power of Expansion: Understanding (a - b)² = a² - 2ab + b²

The equation (a - b)² = a² - 2ab + b² is a fundamental concept in algebra, often referred to as the square of a binomial difference. It's used extensively in various mathematical fields, from simplifying expressions to solving equations.

The Essence of Expansion

The equation itself describes the expansion of the square of a binomial difference. Let's break it down:

  • (a - b)²: This represents squaring the entire expression (a - b). In other words, multiplying it by itself: (a - b) * (a - b)
  • a² - 2ab + b²: This is the expanded form of the square, revealing the individual terms after the multiplication.

Understanding the Expansion

To grasp the equation, we can visualize the process of expanding (a - b)²:

  1. FOIL Method: The FOIL method (First, Outer, Inner, Last) is a handy tool for multiplying binomials. Applying it here, we get:

    • First: a * a = a²
    • Outer: a * (-b) = -ab
    • Inner: (-b) * a = -ab
    • Last: (-b) * (-b) = b²
  2. Combining Like Terms: After multiplying, we combine the 'Outer' and 'Inner' terms, which are both -ab. This results in the final expansion: a² - 2ab + b²

Application and Importance

The formula (a - b)² = a² - 2ab + b² is essential for various reasons:

  • Simplifying Expressions: This equation enables us to simplify complex algebraic expressions by expanding squares of binomial differences.
  • Solving Equations: Understanding the equation is crucial for solving quadratic equations, which often involve squaring binomials.
  • Factorization: The equation is used in reverse for factoring quadratic expressions. Recognizing the pattern allows us to factorize them into the form (a - b)².

Conclusion

The equation (a - b)² = a² - 2ab + b² is a fundamental algebraic concept with wide-ranging applications. Understanding its derivation and significance is crucial for simplifying expressions, solving equations, and factoring polynomials. By mastering this principle, you gain a solid foundation for exploring advanced algebraic concepts.

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