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(2x+1/3)^2-(x-1/2)^2

2 min read Jun 16, 2024
(2x+1/3)^2-(x-1/2)^2

Simplifying the Expression: (2x + 1/3)² - (x - 1/2)²

This article will guide you through the process of simplifying the algebraic expression: (2x + 1/3)² - (x - 1/2)². We'll use the difference of squares pattern and step-by-step simplification to arrive at the final answer.

Understanding the Difference of Squares Pattern

The difference of squares pattern is a fundamental algebraic concept that states:

a² - b² = (a + b)(a - b)

This pattern allows us to factor expressions where two perfect squares are being subtracted.

Applying the Pattern to Our Expression

  1. Identify the squares: In our expression, (2x + 1/3)² and (x - 1/2)² are both perfect squares.

  2. Apply the pattern: We can directly apply the difference of squares pattern:

    (2x + 1/3)² - (x - 1/2)² = [(2x + 1/3) + (x - 1/2)][(2x + 1/3) - (x - 1/2)]

  3. Simplify the expressions inside the brackets:

    [(2x + 1/3) + (x - 1/2)] = 3x - 1/6 [(2x + 1/3) - (x - 1/2)] = x + 5/6

  4. Multiply the simplified expressions:

    (3x - 1/6)(x + 5/6) = 3x² + (29/6)x - 5/36

Conclusion

Therefore, the simplified form of the expression (2x + 1/3)² - (x - 1/2)² is 3x² + (29/6)x - 5/36. By applying the difference of squares pattern, we were able to factor the expression and simplify it into a more manageable form.

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