(x+1)^2-2(2x-1)(1+x)+4x^2-4x+1

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (x+1)^2 - 2(2x-1)(1+x) + 4x^2 - 4x + 1

This article explores the simplification of the given algebraic expression: (x+1)^2 - 2(2x-1)(1+x) + 4x^2 - 4x + 1.

Expanding the Expression

First, we can expand the squared terms and the product of binomials:

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

Now, let's substitute these expansions back into the original expression:

(x^2 + 2x + 1) - 2(2x^2 + x - 1) + 4x^2 - 4x + 1

Simplifying by Distribution and Combining Like Terms

Next, distribute the -2:

x^2 + 2x + 1 - 4x^2 - 2x + 2 + 4x^2 - 4x + 1

Finally, combine like terms:

(x^2 - 4x^2 + 4x^2) + (2x - 2x - 4x) + (1 + 2 + 1)

This simplifies to:

x^2 - 4x + 4

Conclusion

The simplified form of the expression (x+1)^2 - 2(2x-1)(1+x) + 4x^2 - 4x + 1 is x^2 - 4x + 4.

This simplified expression can be further factored as (x-2)^2. This demonstrates the usefulness of algebraic simplification in reducing complex expressions to their simplest forms, often revealing valuable insights into their structure and properties.