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The Difference of Squares: Exploring (100-x)(100+x)
The expression (100-x)(100+x) represents a common mathematical pattern known as the difference of squares. This pattern is a useful shortcut in algebra and can help simplify expressions and solve equations.
Understanding the Pattern
The difference of squares pattern states that:
(a - b)(a + b) = a² - b²
In our case, 'a' is 100 and 'b' is 'x'.
Expanding the Expression
Let's expand (100-x)(100+x) using the distributive property or the FOIL method:
- First: 100 * 100 = 10000
- Outer: 100 * x = 100x
- Inner: -x * 100 = -100x
- Last: -x * x = -x²
Combining like terms, we get: 10000 + 100x - 100x - x² = 10000 - x²
The Result
As you can see, expanding the expression (100-x)(100+x) directly leads to the difference of squares pattern: 100² - x²
Applications
The difference of squares pattern is widely used in:
- Factoring expressions: It can help factorize expressions that have the form of a² - b² into (a-b)(a+b).
- Solving equations: This pattern can be applied to solve equations where the expression is in the form of a² - b² = 0.
- Simplifying expressions: It can simplify complex expressions by recognizing the pattern and replacing it with its equivalent form.
Conclusion
The expression (100-x)(100+x) demonstrates the difference of squares pattern, which provides a useful tool for simplifying expressions, factoring polynomials, and solving equations. Understanding this pattern can streamline your algebraic work and enhance your understanding of mathematical relationships.