(a + b)(a - b)

Algebra-1

1. Fundamental Concepts

Definition: The expression $$(a + b)(a - b)$$ is known as the difference of squares, which simplifies to $$a^2 - b^2$$ .
Like Terms: In this context, like terms refer to the squared terms that result from the expansion of the binomials.
Closure Property: The result of multiplying two binomials of the form $$(a + b)(a - b)$$ always results in a polynomial of degree 2.

2. Key Concepts

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

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 $$(x + 3)(x - 3)$$

Step-by-Step Solution:

Apply the difference of squares formula: $$(x + 3)(x - 3) = x^2 - 3^2$$
Simplify: $$x^2 - 9$$

Validation: Substitute x = 2 → Original: $(2 + 3)(2 - 3) = 5 \cdot (-1) = -5$; Simplified: $2^2 - 9 = 4 - 9 = -5$ ✓

Example 2 (Intermediate)

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

Step-by-Step Solution:

Apply the difference of squares formula: $$(2y + 5)(2y - 5) = (2y)^2 - 5^2$$
Simplify: $$4y^2 - 25$$

Validation: Substitute y = 1 → Original: $(2 \cdot 1 + 5)(2 \cdot 1 - 5) = 7 \cdot (-3) = -21$; Simplified: $4 \cdot 1^2 - 25 = 4 - 25 = -21$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding to distinguish between positive and negative terms.
Error-Proofing: Double-check the signs when applying the difference of squares formula.
Concept Reinforcement: Practice with various values for a and b to reinforce understanding.