%5E2-formula-based-questions.jpeg "(a+b+c)^2 Formula Based Questions")
(a+b+c)^2 Formula: Understanding and Solving Based Questions
The (a+b+c)^2 formula is a fundamental concept in algebra and is widely used in various mathematical problems. In this article, we will delve into the understanding of the (a+b+c)^2 formula, its expansion, and how to solve based questions.
What is the (a+b+c)^2 Formula?
The (a+b+c)^2 formula is a algebraic expression that represents the square of the sum of three terms a, b, and c. It is expressed as:
(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
This formula can be proven by expanding the binomial (a+b+c)^2 using the distributive property of multiplication over addition.
How to Expand the (a+b+c)^2 Formula?
Expanding the (a+b+c)^2 formula involves multiplying the binomial (a+b+c) by itself and simplifying the resulting expression. The step-by-step expansion is as follows:
(a+b+c)^2 = (a+b+c)(a+b+c) = a(a+b+c) + b(a+b+c) + c(a+b+c) = a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca**
Examples of (a+b+c)^2 Formula Based Questions
Here are some examples of questions that can be solved using the (a+b+c)^2 formula:
Example 1: Expand and Simplify
Expand and simplify the expression (x + 2y + 3z)^2.
Solution:
(x + 2y + 3z)^2 = x^2 + (2y)^2 + (3z)^2 + 2x(2y) + 2(2y)(3z) + 2(3z)x = x^2 + 4y^2 + 9z^2 + 4xy + 12yz + 6zx**
Example 2: Find the Value
If a = 2, b = 3, and c = 4, find the value of (a+b+c)^2.
Solution:
(2+3+4)^2 = (2)^2 + (3)^2 + (4)^2 + 2(2)(3) + 2(3)(4) + 2(4)(2) = 4 + 9 + 16 + 12 + 24 + 16 = 81**
Example 3: Simplify the Expression
Simplify the expression (2x + 3y - z)^2.
Solution:
(2x + 3y - z)^2 = (2x)^2 + (3y)^2 + (-z)^2 + 2(2x)(3y) + 2(3y)(-z) + 2(-z)(2x) = 4x^2 + 9y^2 + z^2 + 12xy - 6yz - 4zx**
In conclusion, the (a+b+c)^2 formula is a powerful tool in algebra that helps in expanding and simplifying complex expressions. By understanding the formula and its expansion, we can solve a variety of problems involving the sum of three terms.
