Lecture Notes: Understanding Discriminants in Quadratic Equations

Introduction

• Objective: Understand the concept of discriminants in quadratic equations.
• Quadratic Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
• Discriminant Definition: The term (b^2 - 4ac) in the quadratic formula is called the discriminant.

Importance of Discriminants

• Determines the number and type of solutions for a quadratic equation:
  • Positive Discriminant: Two distinct real roots.
  • Zero Discriminant: One real root (two equal real roots).
  • Negative Discriminant: No real roots (no solutions in real numbers).

Examples

1. Example 1: Quadratic equation (x^2 + 4x + 2)

   • (a = 1, b = 4, c = 2)
   • Discriminant: (b^2 - 4ac = 16 - 8 = 8)
   • Conclusion: Two distinct real roots.

2. Example 2: Quadratic equation with one root

   • Equation touches the x-axis at one point.
   • (a = 1, b = -6, c = 9)
   • Discriminant: (36 - 36 = 0)
   • Conclusion: Two equal real roots (appears as a single point on the graph).

3. Example 3: Quadratic equation with no real roots

   • Graph is above the x-axis.
   • (a = 2, b = -1, c = 3)
   • Discriminant: (1 - 24 = -23)
   • Conclusion: No real roots.

Practice Questions

• Calculate the discriminant for given equations and determine the number of roots.
• Example Calculations:
  • Given (a = 4, b = -4, c = 1): Discriminant = 0, Two equal roots.
  • Given (a = 3, b = -2, c = -5): Discriminant = 64, Two distinct roots.

Advanced Problems

1. Finding Constant Values:

   • Problem: (px^2 + 2x - 3 = 0) has equal roots. Find (p).
   • Approach: (b^2 - 4ac = 0), solve for (p).

2. Complex Quadratic Example:

   • (3qx^2 - 4qx + 4 = 0): Solve for (q) using discriminant condition for equal roots.

Completing the Square & Discriminant Relation

• Concept: Rewriting a quadratic equation in the form (k + p^2 + q).
• Application: Demonstrates how discriminants contribute to determining distinct roots.

Summary

• Understanding the discriminant is essential for determining the nature and number of solutions of quadratic equations.
• Practice identifying (a, b, c) and computing the discriminant to master solving quadratic equations.
• Incorporate the quadratic formula and discriminant knowledge into various problems to strengthen mathematical understanding.