(a-b%2Bc)%3Da%5E2%2Bb%5E2%2Bc%5E2.jpeg "(a+b+c)(a-b+c)=a^2+b^2+c^2")
Understanding the Expansion of (a + b + c)(a - b + c) = a² + b² + c²
This equation presents a fascinating algebraic identity, where the product of two seemingly complex expressions simplifies to a concise and elegant form. Let's break down the steps involved in understanding this equation.
Expansion Using the Distributive Property
The core of this equation lies in applying the distributive property of multiplication. We can expand the left-hand side of the equation step-by-step:
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Distribute (a + b + c) over (a - b + c): (a + b + c)(a - b + c) = a(a - b + c) + b(a - b + c) + c(a - b + c)
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Distribute further: a(a - b + c) + b(a - b + c) + c(a - b + c) = a² - ab + ac + ab - b² + bc + ac - bc + c²
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Combine like terms: a² - ab + ac + ab - b² + bc + ac - bc + c² = a² - b² + c² + 2ac
As you can see, the terms -ab and +ab, +bc and -bc cancel out, leaving us with:
a² - b² + c² + 2ac
The Importance of the Identity
Although the equation initially seems straightforward, it highlights a crucial relationship between algebraic expressions. This particular identity helps simplify complex expressions and can be used to solve various mathematical problems.
For instance, you can utilize this identity to prove other algebraic relationships, or you might find it useful when working with quadratic equations or other higher-order polynomials.
Conclusion
The identity (a + b + c)(a - b + c) = a² + b² + c² is a valuable tool in algebra. By understanding its expansion and the underlying principles, you gain a deeper appreciation for the beauty and elegance of mathematical relationships. This knowledge can empower you to solve various problems and explore further complexities in the world of algebra.