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(2x+1)^2-(x-1)^2=0

2 min read Jun 16, 2024
(2x+1)^2-(x-1)^2=0

Solving the Equation: (2x+1)^2 - (x-1)^2 = 0

This article will guide you through solving the equation (2x+1)^2 - (x-1)^2 = 0. We'll explore various methods and explain the steps involved.

Method 1: Using the Difference of Squares

The equation is in the form of a² - b² = 0, which is a classic difference of squares pattern. Let's break it down:

  1. Identify 'a' and 'b':

    • a = 2x+1
    • b = x-1

  2. Apply the difference of squares formula:

    • (a + b)(a - b) = 0

  3. Substitute 'a' and 'b':

    • (2x+1 + x-1)(2x+1 - x+1) = 0

  4. Simplify:

    • (3x)(x+2) = 0

  5. Solve for x:

    • 3x = 0 or x+2 = 0
    • x = 0 or x = -2

Method 2: Expanding and Simplifying

We can also solve the equation by expanding and simplifying:

  1. Expand the squares:

    • (2x+1)(2x+1) - (x-1)(x-1) = 0
    • 4x² + 4x + 1 - (x² - 2x + 1) = 0

  2. Simplify:

    • 4x² + 4x + 1 - x² + 2x - 1 = 0
    • 3x² + 6x = 0

  3. Factor out 3x:

    • 3x(x + 2) = 0

  4. Solve for x:

    • 3x = 0 or x+2 = 0
    • x = 0 or x = -2

Conclusion

Both methods lead to the same solutions: x = 0 and x = -2. Using the difference of squares pattern can often be a quicker and more efficient method for solving equations of this type. However, expanding and simplifying can be a helpful approach when the difference of squares pattern is not immediately apparent.

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