Binomial squared (subtraction）

Algebra-1

1. Fundamental Concepts

Definition: A binomial squared with subtraction is an expression of the form $$(x - y)^2$$ , which expands to $$x^2 - 2xy + y^2$$ .
Like Terms: Terms that contain the same variables raised to the same powers.
Closure Property: The result of squaring a binomial is always a polynomial.

2. Key Concepts

Basic Rule: $$(x - y)^2 = x^2 - 2xy + y^2$$

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: $$(x - 3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2$$
Simplify: $$x^2 - 6x + 9$$

Validation: Substitute x = 1 → Original: $(1 - 3)^2 = 4$; Simplified: 1 - 6 + 9 = 4 ✓

Example 2 (Intermediate)

Problem: Simplify $$(2y - 5)^2$$

Step-by-Step Solution:

Apply the formula: $$(2y - 5)^2 = (2y)^2 - 2 \cdot 2y \cdot 5 + 5^2$$
Simplify: $$4y^2 - 20y + 25$$

Validation: Substitute y = 1 → Original: $(2 \cdot 1 - 5)^2 = 9$; Simplified: $4 \cdot 1^2 - 20 \cdot 1 + 25 = 9$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish different terms in the expansion.
Error-Proofing: Double-check each term by substituting values for variables.
Concept Reinforcement: Practice with various binomials to reinforce understanding of the pattern.