Lecture Notes: Solving Quadratic Equations

Key Concepts

• Quadratic Equations: Equations of the form ax^2 + bx + c = 0.
• Factoring: Expressing the equation as a product of factors.
• Zero Product Property: If the product of two factors is zero, at least one of the factors must be zero.

Solving Quadratics by Factoring

• Example: A quadratic equation with large coefficients (e.g., 899).
• Objective: Find two numbers that multiply to the constant term and add to the coefficient of the linear term.

Steps to Factor Quadratics

1. Write the Equation in Factored Form: ( (x - m)(x - n) = 0 ).
   • Use minus signs to simplify reading off solutions.
2. Identify Numbers (m, n):
   • Numbers must multiply to the constant term (e.g., 899) and add to the linear coefficient (e.g., 60).

Using the Difference of Squares

• Approach:
  • Consider sums first if multiplication is complex.
  • Solve: ( (30 - u)(30 + u) = 899 ).
  • Difference of Squares: ( a^2 - b^2 = (a-b)(a+b) ), removes middle terms.

Solving Quadratics When Coefficient is Not 1

• Adjust the Coefficient:
  • Divide the entire equation by the leading coefficient to simplify.

Example

• Equation: ( 2x^2 - 16x + 26 = 0 ).
  • Divide by 2: ( x^2 - 8x + 13 = 0 ).
• Un-add the Coefficient:
  • Use a method similar to the difference of squares.
  • Solve: ( (4 - u)(4 + u) = 13 ).
  • Find ( u ) such that: ( u^2 = 3 ).

Historical Context

• Babylonian Mathematics:
  • Used base 60, making multiplication tables cumbersome.
  • Preferred ‘unsumming’ over ‘unmultiplying’ due to this.

Summary

• General Method:
  • Use the difference of squares to simplify factoring.
  • Avoids extensive memorization of specific formulas.
• Philosophy:
  • Mathematics should be accessible.
  • Use intuitive methods when memory or standard methods fail.

Closing

• This method works for all quadratics and does not require memorizing specific formulas. Understanding these concepts can help solve any quadratic equation intuitively.