Binomial squared (addition) - Algebra-1

1. Fundamental Concepts

Definition: A binomial squared (addition) refers to the square of a sum of two terms, expressed as $(a+b)^2$
Expansion Formula: The expansion of $$(a + b)^2$$ is $$a^2 + 2ab + b^2$$ .
Special Case: When squaring a binomial, the middle term is always twice the product of the two terms.

2. Key Concepts

Basic Rule: $$(a + b)^2 = a^2 + 2ab + b^2$$

Degree Preservation: The highest degree in the result matches input

Application: Used to simplify expressions and solve equations in algebra

3. Examples

Example 1 (Basic)

Problem: Simplify $$(x + 3)^2$$

Step-by-Step Solution:

Apply the formula: $$a^2 + 2ab + b^2$$ where $$a = x$$ and $$b = 3$$
Substitute values: $$x^2 + 2(x)(3) + 3^2$$
Simplify: $$x^2 + 6x + 9$$

Validation: Substitute $$x = 1$$ → Original: $$(1 + 3)^2 = 16$$ ; Simplified: $$1^2 + 6(1) + 9 = 16$$ ✓

Example 2 (Intermediate)

Calculate $(3x + 2y)^2$
Solution: Let $a = 3x$ and $b = 2y$, then substitute into the formula$(3x + 2y)^2 = (3x)^2 + 2\times3x\times2y + (2y)^2 = 9x^2 + 12xy + 4y^2$.

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish between different parts of the binomial.
Error-Proofing: Double-check each step by substituting values for variables.
Concept Reinforcement: Practice with various types of binomials to reinforce understanding.