%5E2-formula-derivation.jpeg "(a+b+c)^2 Formula Derivation")
Derivation of the (a + b + c)^2 Formula
The formula for squaring a trinomial (a + b + c) is a useful tool in algebra. It allows us to expand the expression without having to manually multiply it out. Here's how we derive the formula:
Understanding the Concept
The key is to remember that squaring an expression means multiplying it by itself:
(a + b + c)^2 = (a + b + c)(a + b + c)
Expanding the Expression
To expand this, we use the distributive property (also known as FOIL for binomials). This means multiplying each term in the first set of parentheses by each term in the second set of parentheses:
(a + b + c)(a + b + c) = a(a + b + c) + b(a + b + c) + c(a + b + c)
Now, distribute each term:
= a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2
Combining Like Terms
Finally, combine the like terms:
= a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
This is the formula for (a + b + c)^2.
Example
Let's say we want to expand (2x + 3y + 1)^2. We can use the formula:
(2x + 3y + 1)^2 = (2x)^2 + (3y)^2 + (1)^2 + 2(2x)(3y) + 2(2x)(1) + 2(3y)(1)
Simplifying:
= 4x^2 + 9y^2 + 1 + 12xy + 4x + 6y
Conclusion
The (a + b + c)^2 formula provides a shortcut for expanding trinomials. It saves time and effort, especially when dealing with complex expressions. Understanding its derivation helps us appreciate the beauty and logic behind this fundamental algebraic formula.