2%3Da2-2ab%2Bb2-answer.jpeg "(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:
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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
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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.