Quadratic Equations

Overview

• Quadratic Equation: An equation where the highest power of the variable is squared (x²).
• Non-Quadratic Equation: Does not include x² or x² terms cancel out.

Identifying Quadratic Equations

1. Check for x² Term: Must not cancel out with another x².
2. Example of Misidentification:
   • Equation: 4 + x² - x² = 3x
   • Simplified: 4 = 3x (Not quadratic as x² terms cancel out).

Solving Quadratic Equations

1. Move Everything to One Side:
   • Convert to form ax² + bx + c = 0.
   • Example: x² - 8x - 9 = 0
2. Factorize the Equation:
   • Identify type of polynomial (e.g., trinomials, difference of squares, common factors).
   • Example: x² - 8x - 9 is a trinomial, factorizes to (x - 9)(x + 1) = 0
3. Solve for x:
   • Set each factor to zero.
   • Example: x - 9 = 0 or x + 1 = 0
   • Solutions: x = 9 and x = -1

Examples

1. Example 1: Trinomial

   • Equation: x² - 3x - 4 = 0
   • Factorize: (x - 4)(x + 1) = 0
   • Solutions: x = 4, x = -1

2. Example 2: Difference of Squares

   • Equation: x² - 4 = 0
   • Factorize: (x - 2)(x + 2) = 0
   • Solutions: x = 2, x = -2

3. Example 3: Trinomial

   • Equation: x² - 8x + 1 = 0
   • Factorize: (x - 8)(x + 1) = 0
   • Solutions: x = 8, x = -1

4. Example 4: Common Factor

   • Equation: x² - 3x = 0
   • Factorize: x(x - 3) = 0
   • Solutions: x = 0, x = 3

Steps Recap

1. Ensure x² term remains.
2. Move terms to one side (ax² + bx + c = 0).
3. Factorize based on polynomial type.
4. Solve each factor for x.

Key Points

• Quadratic if x² term does not cancel out.
• Zero on one side is crucial (ax² + bx + c = 0).
• Factorize using appropriate method.
  • Trinomial
  • Difference of squares
  • Common factor
• Solve for x from factors.