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Factoring the Difference of Squares: (x-9)(x+9)
The expression (x-9)(x+9) represents a special case of factoring known as the difference of squares. Let's break down what this means and how to apply it.
Understanding the Difference of Squares
The difference of squares pattern is a fundamental concept in algebra. It states that the difference between two perfect squares can be factored into the product of the sum and difference of the square roots of those squares.
In mathematical terms:
a² - b² = (a + b)(a - b)
Applying the Pattern to (x-9)(x+9)
In our expression (x-9)(x+9), we can see that:
- x² is a perfect square (the square of x)
- 9² is a perfect square (the square of 9)
Therefore, we can apply the difference of squares pattern to factor the expression:
(x-9)(x+9) = x² - 9²
Now, we can directly apply the difference of squares formula:
x² - 9² = (x + 9)(x - 9)
Conclusion
By understanding the difference of squares pattern, we can easily factor expressions like (x-9)(x+9). This technique is valuable for simplifying algebraic expressions and solving equations.