Factor Quadratic Trinomials

Algebra-2

1. Fundamental Concepts

Definition: Quadratic trinomials are polynomials of the form $$ax^2 + bx + c$$ where $$a \neq 0$$ .
Factoring: The process of writing a quadratic trinomial as a product of two binomials.
Leading Coefficient: The coefficient of the highest degree term, which is $$a$$ in $$ax^2 + bx + c$$ .

2. Key Concepts

Basic Rule: $$ax^2 + bx + c = (px + q)(rx + s)$$

Product-Sum Method: Find two numbers that multiply to $$ac$$ and add to $$b$$ .

Application: Used to solve quadratic equations and analyze functions.

3. Examples

Example 1 (Basic)

Problem: Factor $$x^2 + 5x + 6$$

Step-by-Step Solution:

Identify $$a = 1$$ , $$b = 5$$ , and $$c = 6$$ . Find two numbers that multiply to $$1 \cdot 6 = 6$$ and add to $$5$$ .
The numbers are $$2$$ and $$3$$ .
Write the factorization: $$(x + 2)(x + 3)$$

Example 2 (Intermediate)

Problem: Factor $$2x^2 + 7x + 3$$

Step-by-Step Solution:

Identify $$a = 2$$ , $$b = 7$$ , and $$c = 3$$ . Find two numbers that multiply to $$2 \cdot 3 = 6$$ and add to $$7$$ .
The numbers are $$6$$ and $$1$$ .
Rewrite the middle term using these numbers: $$2x^2 + 6x + x + 3$$
Group terms: $$(2x^2 + 6x) + (x + 3)$$
Factor by grouping: $$2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$$

4. Problem-Solving Techniques

Visual Strategy: Use a grid or box method to organize terms when factoring complex expressions.
Error-Proofing: Double-check the product and sum of the inner and outer terms after factoring.
Concept Reinforcement: Practice with various types of quadratic trinomials, including those with leading coefficients other than 1.