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Understanding the (a+b+c)2 Expansion
The formula (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca is a fundamental algebraic expansion used in various mathematical disciplines. It helps simplify complex expressions and solve equations.
The Expansion in Detail
This expansion is based on the principle of multiplying a sum by itself. When you square a trinomial like (a+b+c), you are essentially multiplying it by itself:
(a+b+c)2 = (a+b+c)(a+b+c)
To expand this, you need to distribute each term of the first trinomial to each term of the second trinomial:
- a * (a+b+c): This gives us a2 + ab + ac
- b * (a+b+c): This gives us ab + b2 + bc
- c * (a+b+c): This gives us ac + bc + c2
Combining all the terms, we get:
a2 + ab + ac + ab + b2 + bc + ac + bc + c2
Simplifying by combining like terms, we arrive at the final expansion:
(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Real-World Applications
This formula has numerous practical applications, including:
- Algebraic Simplification: It helps simplify complex expressions involving trinomials.
- Geometry: It can be used to derive formulas for calculating areas and volumes.
- Physics: It is used in various physical equations involving sums of quantities.
- Computer Science: It finds applications in algorithms and data structures.
Example
Let's consider an example to illustrate the use of this formula:
Find the value of (2+3+4)2
Using the formula, we get:
(2+3+4)2 = 22 + 32 + 42 + 2(2)(3) + 2(3)(4) + 2(2)(4)
Simplifying:
= 4 + 9 + 16 + 12 + 24 + 16
= 81
Therefore, (2+3+4)2 = 81
Conclusion
The expansion (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca is a powerful tool in algebra. Its ability to simplify expressions and solve equations makes it an essential formula for understanding various mathematical and scientific concepts. Mastering this expansion can significantly enhance your problem-solving skills in diverse fields.