%5E2-formula-example.jpeg "(a+b+c)^2 Formula Example")
Understanding the (a + b + c)^2 Formula
The formula (a + b + c)^2 is a fundamental concept in algebra and finds its applications in various mathematical and real-world problems. It allows us to quickly expand expressions involving the sum of three terms squared. Let's dive deeper into this formula and understand its usage with examples.
The Formula:
The formula states that:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
This means that squaring the sum of three terms results in the sum of the squares of each term plus twice the product of each pair of terms.
Example 1:
Problem: Expand (x + y + 2)^2
Solution:
Using the formula, we get:
(x + y + 2)^2 = x^2 + y^2 + 2^2 + 2(x)(y) + 2(x)(2) + 2(y)(2)
Simplifying further:
(x + y + 2)^2 = x^2 + y^2 + 4 + 2xy + 4x + 4y
Example 2:
Problem: Calculate the value of (3 + 4 + 5)^2
Solution:
We can directly apply the formula:
(3 + 4 + 5)^2 = 3^2 + 4^2 + 5^2 + 2(3)(4) + 2(3)(5) + 2(4)(5)
Simplifying:
(3 + 4 + 5)^2 = 9 + 16 + 25 + 24 + 30 + 40
(3 + 4 + 5)^2 = 144
Why is this Formula Useful?
The (a + b + c)^2 formula simplifies the process of expanding complex expressions. It avoids the tedious step of multiplying the entire expression by itself, saving time and effort. Additionally, the formula plays a crucial role in various mathematical applications, including:
- Factoring quadratic expressions: Understanding this formula helps in factoring expressions containing three terms squared.
- Solving equations: It can be used to simplify equations involving the square of a sum of three terms.
- Geometry: The formula finds its application in deriving formulas related to areas and volumes of geometric shapes.
Understanding the (a + b + c)^2 formula is essential for anyone working with algebraic expressions and can be a valuable tool in various mathematical and scientific applications.