(x%2B5).jpeg "(x-5)(x+5)")
The Special Product: (x-5)(x+5)
The expression (x-5)(x+5) is a special case of a common algebraic pattern. It represents the product of two binomials that are conjugates of each other. Let's explore why this is special and how to simplify it.
Understanding Conjugates
Two binomials are conjugates if they have the same terms but opposite signs in the middle. In our case:
- (x - 5) - The first term is 'x' and the second term is '-5'.
- (x + 5) - The first term is 'x' and the second term is '+5'.
The "Difference of Squares" Pattern
When multiplying conjugates, we get a very specific result:
** (a - b)(a + b) = a² - b² **
This is known as the difference of squares pattern. It's a useful shortcut for expanding expressions like (x-5)(x+5).
Applying the Pattern
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Identify 'a' and 'b':
- In our example, 'a' is 'x' and 'b' is '5'.
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Substitute into the pattern:
- (x - 5)(x + 5) = x² - 5²
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Simplify:
- x² - 5² = x² - 25
Conclusion
Therefore, the simplified form of (x-5)(x+5) is x² - 25. This pattern is extremely valuable for simplifying expressions and solving equations in algebra. Remember to keep the "difference of squares" pattern in mind whenever you encounter conjugates in your math problems.