(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca

Jasonbradley

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Jun 16, 2024

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The Square of a Trinomial: Understanding (a+b+c)^2

The formula (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca is a fundamental algebraic identity that helps us expand the square of a trinomial. Understanding this identity is crucial in various mathematical applications, particularly in algebra and calculus.

Expanding the Square

The formula can be derived by applying the distributive property of multiplication. Let's break it down:

1. Start with (a+b+c)^2: This represents the square of the trinomial (a+b+c).
2. Expand the square: We can rewrite it as (a+b+c) * (a+b+c).
3. Apply the distributive property: We multiply each term in the first trinomial by each term in the second trinomial.

This results in the following expansion:

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

4. Distribute again:

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

5. Combine like terms:

= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

This demonstrates how the formula (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca is derived.

Applications and Significance

The square of a trinomial formula has various applications, including:

• Simplifying algebraic expressions: It helps simplify expressions involving squares of trinomials.
• Solving equations: It can be used to solve equations that involve squares of trinomials.
• Factoring expressions: It can be used to factor expressions that involve squares of trinomials.
• Geometric applications: It can be used to solve problems involving areas and volumes of geometric shapes.

Example

Let's apply the formula to a practical example:

Problem: Expand (2x + 3y + 5)^2

Solution: Using the formula, we get:

(2x + 3y + 5)^2 = (2x)^2 + (3y)^2 + (5)^2 + 2(2x)(3y) + 2(2x)(5) + 2(3y)(5)

Simplifying the expression:

= 4x^2 + 9y^2 + 25 + 12xy + 20x + 30y

Therefore, (2x + 3y + 5)^2 expands to 4x^2 + 9y^2 + 25 + 12xy + 20x + 30y.

Conclusion

The square of a trinomial formula is an essential tool in algebra and other mathematical disciplines. Understanding its derivation and applications allows for efficient simplification, problem-solving, and a deeper understanding of mathematical concepts. By applying this formula, we can simplify complex expressions and solve various problems involving trinomials.