Quadratic Equations Fundamentals

Definition and Forms

• A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable (usually x) is two.
• The general form of a quadratic equation is: ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Types of Quadratic Equations

• Monic quadratic equation: a = 1, e.g., x^2 + bx + c = 0
• Non-monic quadratic equation: a ≠ 1, e.g., 2x^2 + bx + c = 0

Solutions of Quadratic Equations

• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to find the roots of the equation.
• Factoring: if the equation can be written in the form (x - r)(x - s) = 0, then the roots are x = r and x = s.

Graphs of Quadratic Equations

• Parabola: the graph of a quadratic equation, opening upward or downward.
• Vertex: the minimum or maximum point of the parabola.
• Axis of Symmetry: the vertical line passing through the vertex.

Key Features

• Discriminant: b^2 - 4ac, determines the nature of the roots:
  • Positive: two distinct real roots
  • Zero: one repeated real root
  • Negative: two complex conjugate roots
• x-intercepts: the roots of the equation, where the graph crosses the x-axis.