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(a+b+c)^2: Understanding the Expansion of a Binomial Cube
In algebra, expanding the square of a binomial expression, such as (a+b+c)^2, is an essential skill that can be applied to various mathematical problems. In this article, we will delve into the expansion of (a+b+c)^2, explain the concept, and provide examples to illustrate the process.
What is (a+b+c)^2?
The expression (a+b+c)^2 represents the square of the sum of three variables: a, b, and c. In other words, it is the result of multiplying the sum of a, b, and c by itself.
Expanding (a+b+c)^2
To expand (a+b+c)^2, we need to follow the binomial theorem, which states that:
(a+b+c)^2 = (a+b+c)(a+b+c)
Using the distributive property, we can expand the expression as follows:
(a+b+c)(a+b+c) = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
This expansion can be further simplified by combining like terms:
(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
Examples and Applications
Example 1: Expanding (x+y+z)^2
Let's expand (x+y+z)^2 using the formula above:
(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + xz)
Example 2: Finding the Area of a Triangle
The area of a triangle with sides a, b, and c can be found using Heron's formula, which involves expanding (a+b+c)^2:
Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
Expanding (a+b+c)^2 in the formula leads to:
Area = √(a^2 + b^2 + c^2 - 2(ab + bc + ca))
Example 3: Solving a Quadratic Equation
Solving quadratic equations often involves expanding expressions like (a+b+c)^2. For instance, consider the equation:
(x+2+3)^2 = 25
Expanding the left-hand side, we get:
x^2 + 4x + 12 + 4x + 8 + 6 = 25
Simplifying the equation, we can solve for x.
Conclusion
In conclusion, expanding (a+b+c)^2 is a fundamental concept in algebra that has various applications in mathematics, such as finding the area of a triangle, solving quadratic equations, and more. By understanding the expansion formula and its simplification, you can tackle more complex problems with ease.
