(a+3b)^2-(a-3b)^2

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying (a + 3b)^2 - (a - 3b)^2

This expression involves the difference of two squares, a common pattern in algebra. We can simplify it using the following steps:

Understanding the Difference of Squares Pattern

The difference of squares pattern states: a² - b² = (a + b)(a - b)

Applying the Pattern to our Expression

1. Identify a and b: In our expression, (a + 3b)² - (a - 3b)², we can see that:

   • a = (a + 3b)
   • b = (a - 3b)

2. Substitute into the pattern: Using the difference of squares pattern, we can rewrite the expression: [(a + 3b) + (a - 3b)][(a + 3b) - (a - 3b)]

3. Simplify:

   • [(a + 3b) + (a - 3b)] = 2a
   • [(a + 3b) - (a - 3b)] = 6b

4. Final Result: Therefore, (a + 3b)² - (a - 3b)² simplifies to: 2a * 6b = 12ab

Conclusion

By recognizing and applying the difference of squares pattern, we were able to simplify the expression (a + 3b)² - (a - 3b)² to 12ab. This demonstrates how understanding algebraic patterns can significantly simplify complex expressions.