(a+b+c)^2 Formula Wiki

3 min read Jun 16, 2024

(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.