Lecture Notes: Quadratic Equations

Introduction

• Topic: Understanding quadratic equations and their standard form.
• Key focus: Identifying coefficients (A, B, C) in quadratic equations.

Definition of Quadratic Equation

• A quadratic equation is a mathematical statement with a degree of 2.
• Standard form:
  • General Form: ax² + bx + c = 0
  • Where A, B, and C are real numbers, and A ≠ 0.

Components of Quadratic Equations

• Quadratic Term: ax²
• Linear Term: bx
• Constant Term: c

Identifying A, B, C Values

Example 1:

• Given: x² - 5x + 3 = 0
  • A = 1
  • B = -5
  • C = 3

Example 2:

• Given: 9x² - 25 = 0
  • A = 9
  • B = 0
  • C = -25

Example 3:

• Given: 7x² = (1/3)x
  • Rearranged to: 7x² - (1/3)x = 0
  • A = 7
  • B = -1/3
  • C = 0

Writing Quadratic Equations into Standard Form

Example 4:

• Given: x² + x = 4
  • Rearranged to: x² + x - 4 = 0
  • A = 1
  • B = 1
  • C = -4

Example 5:

• Given: 6x² = 9
  • Rearranged to: 6x² - 9 = 0
  • A = 6
  • B = 0
  • C = -9

Example 6:

• Given: -8x² + x = 6
  • Rearranged to: -8x² + x - 6 = 0
  • Changing signs for standard form:
    • A = 8
    • B = -1
    • C = 6

More Complex Examples

Example 7:

• Given: x(x - 2) = 10
  • After expanding: 3x² - 6x - 10 = 0
  • A = 3
  • B = -6
  • C = -10

Example 8:

• Given: (2x + 5)(x - 1) = 6
  • After expanding and simplifying: 2x² + 3x + 1 = 0
  • A = 2
  • B = 3
  • C = 1

Conditions for Quadratic Equations

• If A = 0, it becomes a linear equation.
• B and C can be zero without losing the quadratic characteristic as long as A ≠ 0.

Conclusion

• Reminder: Always write in the standard form of a quadratic equation: ax² + bx + c = 0.
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