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Understanding the (x-y)^2 = x^2 + y^2 - 2xy Formula
The equation (x-y)^2 = x^2 + y^2 - 2xy is a fundamental algebraic identity used in various mathematical contexts. This article aims to explore its derivation, applications, and significance.
Derivation of the Formula
The identity can be derived using the distributive property of multiplication:
(x-y)^2 = (x-y)(x-y)
Expanding the right-hand side, we get:
(x-y)(x-y) = x(x-y) - y(x-y)
Applying the distributive property again:
x(x-y) - y(x-y) = x^2 - xy - xy + y^2
Combining like terms:
x^2 - xy - xy + y^2 = x^2 + y^2 - 2xy
Therefore, (x-y)^2 = x^2 + y^2 - 2xy
Applications of the Formula
This formula has wide-ranging applications in mathematics, including:
- Simplifying Algebraic Expressions: The identity can be used to simplify expressions involving squares of binomials. For instance, (2a - 3b)^2 can be expanded using the formula as 4a^2 - 12ab + 9b^2.
- Solving Equations: The formula can be used to solve equations involving squares of binomials. For example, if x^2 - 6x + 9 = 0, we can recognize the left-hand side as (x-3)^2, making the solution x = 3.
- Geometric Applications: The formula is related to the Pythagorean theorem and can be used to find the length of the diagonal of a rectangle or the distance between two points in a coordinate plane.
- Calculus: The formula plays a role in deriving the chain rule in calculus, which is used to find the derivative of composite functions.
Significance of the Formula
The (x-y)^2 = x^2 + y^2 - 2xy formula is a fundamental algebraic tool that simplifies expressions, solves equations, and connects to various mathematical concepts. It is a cornerstone of algebraic manipulation and its understanding is essential for mastering algebra and its applications.
Conclusion
The (x-y)^2 = x^2 + y^2 - 2xy formula is a powerful algebraic identity with diverse applications in various mathematical fields. Understanding its derivation and applications is essential for developing a strong foundation in mathematics.