%5E2-formula-derivation.jpeg "(a+b+c)^2 Formula Derivation")
(a+b+c)^2 Formula Derivation
The formula for (a+b+c)^2 is a fundamental concept in algebra and is widely used in various mathematical disciplines. In this article, we will derive the formula for (a+b+c)^2 using the binomial theorem.
The Binomial Theorem
The binomial theorem states that for any positive integer n, the following equation holds:
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
Deriving the Formula for (a+b+c)^2
To derive the formula for (a+b+c)^2, we can start by using the binomial theorem with n=2. We get:
$(a+b+c)^2 = (a+b)^2 + 2(a+b)c + c^2$
Now, we can apply the binomial theorem again to expand the (a+b)^2 term:
$(a+b)^2 = a^2 + 2ab + b^2$
Substituting this back into the original equation, we get:
$(a+b+c)^2 = (a^2 + 2ab + b^2) + 2(a+b)c + c^2$
Simplifying the Expression
We can simplify the expression by combining like terms:
$(a+b+c)^2 = a^2 + 2ab + b^2 + 2ac + 2bc + c^2$
This is the final formula for (a+b+c)^2. It is a quadratic expression in three variables, and it has numerous applications in mathematics, physics, and engineering.
Conclusion
In this article, we have derived the formula for (a+b+c)^2 using the binomial theorem. The formula is a fundamental tool in algebra and is widely used in various mathematical disciplines. It has numerous applications in mathematics, physics, and engineering, and it is an essential concept for anyone studying mathematics or science.
