Quadratic Functions

Introduction

• A quadratic function is expressed in the form: ( f(x) = ax^2 + bx + c ).
• Graphically represented as a parabola on a coordinate plane.

Solutions to Quadratic Equations

• When set equal to zero ( ax^2 + bx + c = 0 ), it determines where the parabola crosses the x-axis.
• Types of Solutions:
  • Two real solutions: The parabola crosses the x-axis at two points.
  • One real solution: The parabola touches the x-axis at one point (vertex).
  • No real solutions: The parabola does not cross the x-axis; complex solutions can be found using the quadratic formula.

Example Quadratic Function

• Example: ( f(x) = x^2 + x - 2 ).
• Solutions are the x-intercepts: (-2, 0) and (1, 0).

Methods to Find Solutions

• Graphing: Visual method to locate where the parabola crosses the x-axis.
• Factoring: Algebraic method to break down the equation.
• Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), used when factoring is difficult.

Advanced Methods

• Completing the Square: Used for quadratic equations that are not easily factored.

Real and Complex Solutions

• Complex solutions occur when the parabola doesn’t intersect the x-axis.
• Complex solutions have real and imaginary parts, solvable using the quadratic formula.

Summary

• Quadratic equations have been studied for centuries with various methods to solve them.
• Understanding how quadratics work is crucial for solving these equations effectively.