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(2x+1)^2-(2x-1)^2

2 min read Jun 16, 2024
(2x+1)^2-(2x-1)^2

Expanding and Simplifying (2x+1)^2 - (2x-1)^2

This expression involves squaring binomials and then subtracting them. Let's break it down step by step.

Expanding the Squares

  • (2x+1)^2: This represents the product of (2x+1) with itself. We can expand it using the FOIL method (First, Outer, Inner, Last) or by recognizing the pattern of a squared binomial:

    • FOIL: (2x+1)(2x+1) = 4x² + 2x + 2x + 1 = 4x² + 4x + 1
    • Pattern: (a+b)² = a² + 2ab + b²
      • Applying the pattern: (2x+1)² = (2x)² + 2(2x)(1) + (1)² = 4x² + 4x + 1

  • (2x-1)^2: Similarly, we can expand this using FOIL or the pattern:

    • FOIL: (2x-1)(2x-1) = 4x² - 2x - 2x + 1 = 4x² - 4x + 1
    • Pattern: (a-b)² = a² - 2ab + b²
      • Applying the pattern: (2x-1)² = (2x)² - 2(2x)(1) + (-1)² = 4x² - 4x + 1

Subtracting the Expanded Expressions

Now, we have:

(2x+1)² - (2x-1)² = (4x² + 4x + 1) - (4x² - 4x + 1)

To subtract, we distribute the negative sign:

4x² + 4x + 1 - 4x² + 4x - 1

Simplifying the Expression

Notice that the 4x² and -4x² terms cancel out. Also, the 1 and -1 terms cancel out. This leaves us with:

4x + 4x = 8x

Conclusion

Therefore, the simplified form of (2x+1)² - (2x-1)² is 8x. This illustrates the power of recognizing patterns and using algebraic manipulations to simplify complex expressions.

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