Quadratic Equations

Jul 12, 2024

Quadratic Equations Lecture

Introduction

  • Title: "Hi Nathan Inaba and Quadratic Equation"
  • Video by: Hyman Akiyama
  • Objective: Understand and solve quadratic equations in standard form (ax² + bx + c = 0)
  • Topics Covered:
    • Identifying quadratic equations
    • Identifying values of a, b, and c

Definitions

Quadratic Equation

  • Quadratic Equation: A mathematical sentence of degree 2 with the highest exponent being 2.
  • Standard Form: ax² + bx + c = 0, where:
    • a, b, c are real numbers
    • a ≠ 0 (to ensure it remains quadratic)

Terms in Quadratic Equation

  1. Quadratic Term (ax²): Term with x²
  2. Linear Term (bx): Term with x
  3. Constant Term (c): Term without x

Identifying a, b, and c

Examples

  1. x² - 5x + 3 = 0
    • a = 1 (coefficient of x²)
    • b = -5 (coefficient of x)
    • c = 3 (constant term)
  2. 4m² + 4m + 1 = 0
    • a = 4
    • b = 4
    • c = 1
  3. 9r² - 25 = 0
    • a = 9
    • b = 0 (no x term)
    • c = -25
  4. 1/2x² + 3
    • a = 1/2
    • b = 3
    • c = 0 (no constant term)

Writing Equations in Standard Form

  • Example 1: x² + x = 4
    • Transform: x² + x - 4 = 0
    • a = 1, b = 1, c = -4
  • Example 2: 7x² = 1/3 x
    • Transform: 7x² - 1/3x = 0
    • a = 7, b = -1/3, c = 0
  • Example 3: 6x² = 9
    • Transform: 6x² - 9 = 0
    • a = 6, b = 0, c = -9
  • Example 4: -8x² + x = 6
    • Transform: -8x² + x - 6 = 0
    • Multiply by -1: 8x² - x + 6 = 0
    • a = 8, b = -1, c = 6

Advanced Examples

Solving Binomials

  • Example 1: 3x(x - 2) = 10
    • Expand: 3x² - 6x = 10
    • Transform: 3x² - 6x - 10 = 0
    • a = 3, b = -6, c = -10
  • Example 2: (2x + 5)(x - 1) = 6
    • Expand: 2x² - 2x + 5x - 5 = 6
    • Combine like terms: 2x² + 3x - 5
    • Transform: 2x² + 3x - 5 + 6 = 0
    • Simplify: 2x² + 3x + 1 = 0
    • a = 2, b = 3, c = 1

Key Points

  • Always ensure equation is in standard form: ax² + bx + c = 0
  • a should never be 0 (or it becomes linear)
  • b and c can be 0 (it remains quadratic)
  • Quadratic Equation is defined by the highest exponent being 2

Conclusion

  • Recap on identifying a, b, and c
  • Transforming equations to standard form
  • Importance of each term

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