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

2 min read Jul 03, 2024

(a+b+c)²=a²+b²+c²+2ab+2bc+2ca

Introduction

In algebra, the expansion of a squared binomial is a fundamental concept. When we square a binomial, we get a quadratic trinomial. In this article, we will discuss the expansion of **(a+b+c)**² and its equivalent form a²+b²+c²+2ab+2bc+2ca.

Expansion of (a+b+c)²

The expansion of **(a+b+c)**² is a quadratic trinomial that can be written as:

**(a+b+c)**² = (a+b+c)(a+b+c)

= a(a+b+c) + b(a+b+c) + c(a+b+c)

= a² + ab + ac + ba + b² + bc + ca + cb + c²

Simplification

Combining like terms, we get:

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

Equivalent Form

The expansion of **(a+b+c)**² is equivalent to:

a² + b² + c² + 2ab + 2bc + 2ca

This form is often used in various mathematical applications, such as calculus, geometry, and algebra.

Example

Let's illustrate this expansion with an example:

Suppose we want to expand **(2+3+4)**².

Using the expansion, we get:

**(2+3+4)**² = 2² + 3² + 4² + 2(2)(3) + 2(3)(4) + 2(2)(4)

= 4 + 9 + 16 + 12 + 24 + 16

= 81

Conclusion

In conclusion, the expansion of **(a+b+c)**² is a quadratic trinomial that can be simplified to a² + b² + c² + 2ab + 2bc + 2ca. This equivalent form is widely used in various mathematical applications and is an essential concept in algebra.