(a + b + c)^2 Formula Explained
The formula (a + b + c)^2 is a fundamental algebraic identity that expands the square of a trinomial. It's a useful tool for simplifying expressions and solving equations.
The Formula
The expanded form of (a + b + c)^2 is:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
Derivation
The formula can be derived by applying the distributive property twice:

First expansion: (a + b + c)^2 = (a + b + c)(a + b + c) = a(a + b + c) + b(a + b + c) + c(a + b + c)

Second expansion: = a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2

Combining like terms: = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
Applications
The (a + b + c)^2 formula has various applications in algebra, geometry, and other fields. Some common uses include:
 Simplifying expressions: The formula can be used to quickly expand and simplify expressions containing squared trinomials.
 Solving equations: The formula can help solve equations involving trinomials by substituting the expanded form.
 Geometry: The formula can be used to derive area and volume formulas for various geometric shapes.
 Calculus: The formula is used in calculus for differentiating and integrating functions involving trinomials.
Examples
Example 1: Expand (x + y + 2)^2
Using the formula:
(x + y + 2)^2 = x^2 + y^2 + 2^2 + 2xy + 2x(2) + 2y(2) = x^2 + y^2 + 4 + 2xy + 4x + 4y
Example 2: Solve the equation (x + 2 + 3)^2 = 25
Using the formula:
x^2 + 2^2 + 3^2 + 2(x)(2) + 2(x)(3) + 2(2)(3) = 25 x^2 + 4 + 9 + 4x + 6x + 12 = 25 x^2 + 10x + 1 = 0
Now you can solve the quadratic equation for x.
Conclusion
The (a + b + c)^2 formula is a powerful tool in algebra and beyond. Understanding its derivation and applications can significantly improve your ability to simplify expressions, solve equations, and tackle more complex problems.