Lecture Notes: Solving Quadratic Equations

Instructor Introduction

• Instructor: Fiore
• Objective: Help determine the most efficient technique for solving quadratic equations.

Overview of Techniques

• A table with three columns:
  1. Description and Form of Quadratic Equation
  2. Most Efficient Solution Method
  3. Example

Detailed Explanation of Quadratic Forms and Solution Methods

1. Standard Quadratic Equation

• Form: (ax^2 + bx + c = 0) and can be easily factored.
• Efficient Method: Factorization using the Zero Product Principle.
• Example:
  • Equation: (3x^2 + 5x - 2 = 0)
  • Factor to: ((3x - 1)(x + 2) = 0)
  • Set each factor to zero and solve:
    • (3x - 1 = 0) leads to (x = \frac{1}{3})
    • (x + 2 = 0) leads to (x = -2)

2. Quadratic Equation without a Constant

• Form: (ax^2 + bx = 0) with (c = 0).
• Efficient Method: Factorization using the Zero Product Principle.
• Example:
  • Factor (6x^2 + 9x = 0) as (3x(2x+3) = 0)
  • Solve: (3x = 0) gives (x = 0), (2x + 3 = 0) gives (x = -\frac{3}{2})

3. Quadratic Without Linear Term

• Form: (ax^2 + c = 0) with (b = 0).
• Efficient Method: Solve by isolating (x^2) and applying the square root property.
• Example:
  • Equation: (7x^2 - 4 = 0) leads to (x^2 = \frac{4}{7})
  • Use square root property: (x = \pm \sqrt{\frac{4}{7}})
  • Rationalize denominator: (x = \pm \frac{2\sqrt{7}}{7})

4. Quadratic in Form (u^2 = d)

• Efficient Method: Square root property.
• Example:
  • Equation: ((x + 4)^2 = 5)
  • Solve by applying square root: (x + 4 = \pm \sqrt{5})
  • Simplify: (x = -4 \pm \sqrt{5})

5. Hard to Factor Quadratics

• Form: (ax^2 + bx + c = 0) where factoring is difficult.
• Efficient Method: Quadratic formula.
• Example:
  • Equation: (x^2 - 2x - 6 = 0)
  • Use formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
  • Calculate: (x = 1 \pm \sqrt{7})

Solving Quadratics by Choice

Example Problem:

• Problem: (x^2 = 4x - 7)
• Re-arrange: (x^2 - 4x + 7 = 0)
• Recognize hard to factor, use quadratic formula.
• Identify (a = 1, b = -4, c = 7), substitute in formula.
• Simplify: (x = 2 \pm i\sqrt{3})

Application Quiz

• Problem: (x^2 - 6x + 13 = 0)
• Options (Incorrect):
  • A) {1, 5}
  • C) {3 \pm 4i}
• Correct Option:
  • B) {3 \pm 2i}
• Solution: Use quadratic formula, (x = 3 \pm 2i)

Key Points:

• Always consider the form of the equation to choose the best method.
• Quadratic formula is versatile for complex solutions.
• Simplification and correct mathematical operations are crucial for accuracy.