(a-b)^2+(b-c)^2+(c-a)^2 Formula

3 min read Jun 16, 2024

Understanding the Formula (a - b)² + (b - c)² + (c - a)²

The formula (a - b)² + (b - c)² + (c - a)² is a useful algebraic expression that has applications in various mathematical and geometric contexts. It is used to represent the sum of squared differences between three variables, 'a', 'b', and 'c'.

Expanding the Formula

The formula can be expanded by applying the algebraic identity: (x - y)² = x² - 2xy + y².

Expanding the formula, we get:

(a - b)² = a² - 2ab + b²
(b - c)² = b² - 2bc + c²
(c - a)² = c² - 2ca + a²

Adding all these together, we get:

(a - b)² + (b - c)² + (c - a)² = 2a² + 2b² + 2c² - 2ab - 2bc - 2ca

Applications

This formula has various applications in different fields, including:

1. Geometry:

Triangle Inequality: The formula can be used to prove the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Distance Formula: The formula relates to the distance formula, which calculates the distance between two points in a coordinate plane.

2. Algebra:

Simplification of expressions: The formula can be used to simplify complex algebraic expressions involving squared differences.
Solving equations: The formula can be helpful in solving equations involving squared differences.

3. Statistics:

Variance Calculation: In statistics, the formula can be used to calculate the variance of a set of data points.

Example

Question: Simplify the expression (2 - 3)² + (3 - 1)² + (1 - 2)²

Solution:

Using the formula:

(2 - 3)² + (3 - 1)² + (1 - 2)² = 2(2²) + 2(3²) + 2(1²) - 2(2)(3) - 2(3)(1) - 2(1)(2)
= 8 + 18 + 2 - 12 - 6 - 4
= 6

Therefore, the simplified value of the expression is 6.

Conclusion

The formula (a - b)² + (b - c)² + (c - a)² is a fundamental algebraic expression with diverse applications in various mathematical and geometric contexts. Understanding its expansion and applications can be beneficial in solving a wide range of problems related to algebra, geometry, and statistics.