Quadratic Equations Lecture Notes

Introduction to Quadratic Equations

• Definition of a quadratic equation: An equation that includes an x squared term (x²).
• Example of a non-quadratic equation: An equation without an x squared term.
• Caution: Some equations may appear quadratic but are not (e.g., if x² cancels out).

Identifying Quadratic Equations

• Example: If x² is moved to one side and cancels out, the equation is not quadratic.
• Quadratic equations must retain the x² term without cancellation.

Steps to Solve a Quadratic Equation

1. Move everything to one side of the equation.

   • Example transformation: 9 becomes -9 on the left side, leading to x² - 8x - 9 = 0.
   • The zero is crucial in quadratic equations.

2. Factor the equation.

   • Several ways to factor: difference of squares, common factors, trinomials.
   • Identify the type of factoring needed based on the equation structure.

3. Find the solutions by setting each factor to zero.

   • Example: From factors x - 9 and x + 1, set to zero to find solutions:
     • x - 9 = 0 leads to x = 9.
     • x + 1 = 0 leads to x = -1.

Example Problems

1. Example 1:

   • Start: x² - 3x - 4 = 0
   • Factor: (x - 4)(x + 1)
   • Solutions: x = 4 or x = -1.

2. Example 2:

   • Already has zero: x² - 4 = 0
   • Recognized as difference of squares: (x - 2)(x + 2)
   • Solutions: x = 2 or x = -2.

3. Example 3:

   • Start: x² - 8x + 1 = 0
   • Factor: (x - 8)(x + 1)
   • Solutions: x = 8 or x = -1.

4. Example 4:

   • Start: x² - 3x = 0
   • Recognized common factor: x(x - 3) = 0
   • Solutions: x = 0 or x = 3.

Conclusion

• Quadratic equations require specific methods for solving, including moving terms, factoring, and finding solutions through setting factors to zero.