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The (x-y)(x+y) Formula: A Powerful Tool for Algebraic Expansion
Introduction
The (x-y)(x+y) formula is a fundamental concept in algebra that helps to expand and simplify complex expressions. It is a powerful tool that can be used to factorize quadratic expressions and solve equations. In this article, we will explore the (x-y)(x+y) formula, its derivation, and some examples of its application.
The Formula
The (x-y)(x+y) formula is given by:
(x-y)(x+y) = x^2 - y^2
This formula can be used to expand the product of two binomials, where one binomial is the difference of two terms and the other binomial is the sum of the same two terms.
Derivation
To derive the (x-y)(x+y) formula, let's start by expanding the product of the two binomials:
(x-y)(x+y) = x(x+y) - y(x+y)
= x^2 + xy - yx - y^2
= x^2 - y^2
As you can see, the middle terms xy and -yx cancel each other out, leaving us with the simplified expression x^2 - y^2.
Examples
The (x-y)(x+y) formula has many applications in algebra and can be used to simplify complex expressions. Here are a few examples:
Example 1
Expand and simplify the expression (2x-3)(2x+3).
Using the (x-y)(x+y) formula, we get:
(2x-3)(2x+3) = (2x)^2 - 3^2
= 4x^2 - 9
Example 2
Solve the equation (x-2)(x+2) = 0.
Using the (x-y)(x+y) formula, we get:
(x-2)(x+2) = x^2 - 2^2
= x^2 - 4
= 0
Solving for x, we get:
x^2 = 4
x = ±2
Therefore, the solutions to the equation are x = 2 and x = -2.
Conclusion
The (x-y)(x+y) formula is a powerful tool that can be used to simplify complex algebraic expressions and solve equations. By understanding how to apply this formula, you can expand your skills in algebra and tackle more challenging problems.
