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

3 min read Jun 16, 2024
(a-b)^2=a^2-b^2

The Misconception: (a - b)^2 = a^2 - b^2

The equation (a - b)^2 = a^2 - b^2 is a common misconception in algebra. While it may seem intuitive at first glance, it is incorrect.

Why It's Wrong

The correct expansion of (a - b)^2 is (a - b)^2 = a^2 - 2ab + b^2.

Let's break it down:

  1. (a - b)^2 represents the square of the binomial (a - b). This means we are multiplying the binomial by itself: (a - b)^2 = (a - b)(a - b)

  2. To expand this, we use the distributive property (also known as FOIL - First, Outer, Inner, Last): (a - b)(a - b) = a(a - b) - b(a - b)

  3. Applying the distributive property again: a(a - b) - b(a - b) = a^2 - ab - ba + b^2

  4. Combining like terms: a^2 - ab - ba + b^2 = a^2 - 2ab + b^2

The Importance of Correct Expansion

Understanding the correct expansion of (a - b)^2 is crucial for various reasons:

  • Accuracy in calculations: Using the incorrect formula will lead to inaccurate results.
  • Solving equations: Many mathematical problems involve squaring binomials. Using the correct expansion is essential for finding the correct solutions.
  • Understanding algebraic concepts: Mastering the expansion of binomials is a fundamental concept in algebra and paves the way for understanding more complex algebraic expressions.

Conclusion

Remember, (a - b)^2 = a^2 - 2ab + b^2, not a^2 - b^2. Pay attention to the middle term (-2ab) and avoid falling into the trap of this common misconception.

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