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What is Quadratic Factorization?
Understanding the fundamentals
Quadratic factorization is the process of expressing a quadratic polynomial of the form ax² + bx + c as the product of two linear polynomials (linear factors). The standard form is typically a(x - r₁)(x - r₂), where r₁ and r₂ are the two roots of the original equation ax² + bx + c = 0.
For example, the quadratic polynomial x² - 5x + 6 can be factored as (x - 2)(x - 3), indicating that x = 2 and x = 3 are the roots of the equation x² - 5x + 6 = 0.
Methods of Quadratic Factorization
Three powerful approaches to solve quadratics
There are multiple methods for polynomial factorization, and different strategies can be chosen based on the type and complexity of the polynomial:
Smart Cross Method
Fast and intuitive approach for most quadratics
Step-by-Step Process:
Identify the coefficients a, b, and c in the quadratic ax² + bx + c.
Find two numbers that multiply to ac and add to b.
Use these numbers to split the middle term and factor by grouping.
Perfect Square Technique
Ideal for completing the square transformations
Step-by-Step Process:
Rewrite the quadratic in the form ax² + bx + c = 0.
Complete the square by adding and subtracting (b/2)².
Factor the perfect square trinomial and simplify.
Advanced Formula Method
Reliable solution using the quadratic formula
Step-by-Step Process:
Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.
Calculate the discriminant D = b² - 4ac to determine the nature of roots.
Express the factored form as a(x - r₁)(x - r₂) using the calculated roots.
Quadratic Factorization Examples
Step-by-step solutions to master the concepts
Here are practical examples that demonstrate different factorization scenarios:
Example 1: Simple Factoring
Problem: Factor x² - 5x + 6
Solution: (x - 2)(x - 3)
We need two numbers that multiply to 6 and add to -5. These are -2 and -3.
Example 2: Leading Coefficient
Problem: Factor 2x² + 7x + 3
Solution: (2x + 1)(x + 3)
Using the ac method: ac = 6, we need factors of 6 that add to 7 (1 and 6).
Example 3: Non-factorable
Problem: Factor x² + x + 1
Result: Cannot be factored over real numbers
Discriminant D < 0, so this quadratic has complex roots.
Example 4: Perfect Square
Problem: Factor x² - 6x + 9
Solution: (x - 3)²
This is a perfect square trinomial: (a - b)² = a² - 2ab + b².
Frequently Asked Questions
Common questions about quadratic factorization
Tip: Relationship Between Discriminant and Factorization
Key insight for understanding solution types
The discriminant D = b² - 4ac determines the factorization possibility: if D > 0, the quadratic has two distinct real roots and can be factored; if D = 0, it has one repeated root (perfect square); if D < 0, it has complex roots and cannot be factored over real numbers.