Factor Difference of Two Squares

Algebra-2

1. Fundamental Concepts

Definition: The difference of two squares is a special case in algebra where an expression takes the form $$a^2 - b^2$$ , which can be factored into $$(a + b)(a - b)$$ .
Identifying Terms: To factor a quadratic expression as a difference of two squares, both terms must be perfect squares and the expression must be a subtraction.
Application: This concept is widely used in simplifying complex expressions and solving equations.

2. Key Concepts

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

Degree Preservation: The highest degree in the result matches input

Application: Used to simplify expressions and solve equations efficiently

3. Examples

Example 1 (Basic)

Problem: Factor the expression $$x^2 - 9$$ .

Step-by-Step Solution:

Identify the form: $$x^2 - 9 = x^2 - 3^2$$
Apply the formula: $$(x + 3)(x - 3)$$

Example 2 (Intermediate)

Problem: Factor the expression $$16y^2 - 25z^2$$ .

Step-by-Step Solution:

Identify the form: $$16y^2 - 25z^2 = (4y)^2 - (5z)^2$$
Apply the formula: $$(4y + 5z)(4y - 5z)$$

4. Problem-Solving Techniques

Pattern Recognition: Look for expressions that fit the form $$a^2 - b^2$$ .
Substitution Method: Use substitution to verify the correctness of the factored form.
Practice with Variety: Practice with different types of expressions to reinforce understanding.