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Factoring the Difference of Squares: (x-3)(x+3)
In mathematics, we often encounter expressions that can be simplified or factored into a more manageable form. One such expression is (x-3)(x+3). This expression exemplifies the difference of squares pattern, a useful tool for factoring certain quadratic expressions.
Understanding the Difference of Squares Pattern
The difference of squares pattern states that the difference of two perfect squares can be factored as:
a² - b² = (a + b)(a - b)
In our expression, (x-3)(x+3), we can identify:
- a = x
- b = 3
Therefore, our expression fits the difference of squares pattern.
Factoring (x-3)(x+3)
Following the difference of squares pattern, we can factor (x-3)(x+3) as:
(x-3)(x+3) = x² - 3²
Simplifying the expression, we get:
x² - 3² = x² - 9
Therefore, the factored form of (x-3)(x+3) is x² - 9.
The Significance of Factoring
Factoring expressions like (x-3)(x+3) is significant for several reasons:
- Simplifying expressions: Factoring can make complex expressions easier to work with, particularly when solving equations or performing further mathematical operations.
- Finding roots: Factoring quadratic equations helps determine their roots (where the equation equals zero), leading to solutions for various problems.
- Understanding relationships: The difference of squares pattern illustrates a fundamental relationship between squares and their factors, enhancing our understanding of mathematical patterns.
In conclusion, understanding and applying the difference of squares pattern is essential for simplifying expressions and solving mathematical problems. By factoring (x-3)(x+3) into x² - 9, we gain a deeper understanding of the relationship between squares and their factors, making us better equipped to tackle more complex mathematical challenges.