Quadratic Equations: Complete Guide

Standard Form

ax² + bx + c = 0

Solution Methods

Factorization

Express as (x - α)(x - β) = 0

Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a

Completing the Square

Convert to (x + p)² = q form

Properties of Roots

• Sum of roots (α + β) = -b/a
• Product of roots (αβ) = c/a
• Difference of roots = √(b² - 4ac) / a

Nature of Roots

• D = b² - 4ac (discriminant)
• D > 0: Real and distinct
• D = 0: Real and equal
• D < 0: Imaginary

Common Patterns

• If α, β are roots of x² + px + q = 0:

- α² + β² = p² - 2q

- α³ + β³ = -p³ + 3pq

- 1/α + 1/β = -p/q

CAT Strategy

• Try factorization first - fastest method
• Use sum/product of roots for relationship questions
• Check answer options by substitution