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The Difference of Squares: (a-1)(a+1)
The expression (a-1)(a+1) is a special case of a common algebraic pattern known as the difference of squares. This pattern is incredibly useful for simplifying expressions and solving equations.
Understanding the Pattern
The difference of squares pattern states that:
(a - b)(a + b) = a² - b²
In our case, a = a and b = 1. Therefore, we can directly apply the pattern:
(a - 1)(a + 1) = a² - 1²
Simplifying the Expression
Simplifying the expression further:
(a - 1)(a + 1) = a² - 1
This is the simplest form of the expression, and it demonstrates the power of recognizing the difference of squares pattern.
Applications
The difference of squares pattern has many applications in mathematics, including:
- Factoring expressions: You can use the pattern to factor expressions that have the form of a² - b².
- Solving equations: You can use the pattern to simplify equations and make them easier to solve.
- Simplifying expressions in calculus: The difference of squares pattern can be used to simplify complex expressions in calculus.
Example
Let's say you have the expression x² - 9. Recognizing this as a difference of squares (where a = x and b = 3), we can factor it:
x² - 9 = (x - 3)(x + 3)
Conclusion
The difference of squares pattern is a fundamental concept in algebra. Understanding and applying this pattern can significantly simplify mathematical expressions and make solving problems more efficient.