Lecture Notes: Quadratic Equations and Factorization

Solving Quadratic Equations

• Methods to Solve
  • Factorization: Use brackets to solve expressions, typically when easy to do so.
  • Quadratic Formula: Works for any trinomial or expression where the highest exponent is 2. Good alternative when factorization seems difficult.

Example Problems

Problem 1: Basic Quadratic

• Equation: x^2 - 3x - 4 = 0
• Factorization:
  • Make brackets: (x - 4)(x + 1) = 0
  • Solutions: x = 4 or x = -1
• Using Formula:
  • Plug values into formula: x = [-b ± √(b² - 4ac)] / 2a
  • Important to use brackets around negative values to avoid errors.
  • Solutions checked: x = 4, x = -1

Problem 2: Factorizing with Zero

• Equation: x^2 - 7x + 10 = 0
• Step by Step:
  • Move terms to one side: x^2 - 7x + 10 = 0
  • Factorization: (x - 5)(x - 2) = 0
  • Solutions: x = 5, x = 2

Problem 3: Complex Quadratic

• Equation: 2x^2 - 7x + 3 = 0
• Using Formula:
  • Plug into formula since factorization is less obvious.
  • Solutions: x = 3, x = 0.5

Problem 4: Direct Formula

• Equation: -x^2 + 11x - 8 = 0
• Solution:
  • Use formula directly.
  • Solutions: x ≈ 0.78, x ≈ 10.22

Problem 5: Special Case - Common Factor

• Equation: x^2 - 3x = 0
• Factorization:
  • Common factor: x(x - 3) = 0
  • Solutions: x = 0, x = 3

Problem 6: Avoiding Common Mistakes

• Equation: x^2 = 4x
• Correct Approach:
  • Move everything to one side: x^2 - 4x = 0
  • Factorization: x(x - 4) = 0
  • Solutions: x = 0, x = 4
  • Warning against canceling terms directly, which can lead to missing solutions.

Problem 7: Complicated Equations

• Approach:
  • Use quadratic formula for complex factorization.
  • Example solution: x ≈ 2.3, x ≈ 0.56

Key Takeaways

• Always Move Terms: When dealing with an x² term, move everything to one side to set the equation to zero.
• Bracket Usage: Use brackets in the quadratic formula to ensure correct computation, especially with negative values.
• Formula Application: The quadratic formula is a reliable tool and can be used even when factorization appears straightforward.
• Check for Common Factors: Don't assume all are trinomials; check for differences of squares or common factors.

Conclusion

• Practice both methods but rely on the formula for consistency across problem types.
• Avoid shortcuts that skip necessary steps, as they can lead to incorrect solutions.