Quadratic Equations Cheat Sheet

Key Definitions

• Quadratic Equation: An equation of the form ax² + bx + c = 0, where a ≠ 0.

• Roots: Solutions of the quadratic equation, found using the quadratic formula.

• Discriminant: D = b² - 4ac; determines the nature of roots.

Important Formulas

• Quadratic Formula:

 
  x = (-b ± √D) / (2a) 
  

• x: roots of the equation

• a: coefficient of x²

• b: coefficient of x

• c: constant term

• D: discriminant

• Sum of Roots:

  α + β = -b/a
  

• Product of Roots:

  αβ = c/a
  

Critical Concepts

• Types of Roots based on Discriminant:

• D > 0: Two distinct real roots

• D = 0: One real root (repeated)

• D < 0: Two complex roots

• Factoring: If possible, express ax² + bx + c as (px + q)(rx + s).

• Graph: Parabola opens upwards if a > 0, downwards if a < 0.

Common Mistakes to Avoid

• Forgetting that a cannot be zero.

• Miscalculating the discriminant.

• Confusing the signs in the quadratic formula.

Quick Memory Tricks/Mnemonics

• "D is for Discriminant": Remember that the discriminant determines the type of roots.

• "Sum and Product": For roots α and β, think Sum = -b/a, Product = c/a.

Key Diagrams

• Parabola Shape:

       |
      |    
       |        *
       |          *
       |_____________
  

• Vertex: Minimum/Maximum point of the parabola.

• Discriminant Visualization:

  D > 0  -->  Two roots
  D = 0  -->  One root
  D < 0  -->  No real roots
  

Tip: Always check the discriminant first to quickly assess the nature of the roots!