Factoring Quadratic Equations with Guess and Check

Apr 30, 2025

Lecture Notes: Guess and Check Method for Factoring

Introduction

  • Speaker: Mr. Lam
  • Topic: Review of the Guess and Check method for factoring quadratic equations, preceded by a refresher on multiplying binomials.

Multiplying Binomials (FOIL Method)

  1. Example: Multiply (3x + 4)(4x - 1)
    • FOIL stands for First, Outer, Inner, Last terms
    • Calculate:
      • First: 3x * 4x = 12x²
      • Outer: 3x * (-1) = -3x
      • Inner: 4 * 4x = 16x
      • Last: 4 * (-1) = -4
    • Combine like terms:
      • 12x² + (-3x + 16x) - 4 = 12x² + 13x - 4
    • Observation: The first number in the binomials gives the first term's coefficient. The last number gives the last term's constant and the middle term is found by combining like terms.

Guess and Check Method

Problem 1: Factor 3x² + 11x + 6

  1. List factors: 3x²
    • Factors: 3, 1
  2. List factors: 6
    • Factors: 1, 6 and 2, 3
  3. Set up binomials:
    • First terms: 3x and 1x
  4. Choose factors for multiplication:
    • Avoid GCF issues (e.g., don't put 6 next to 3, use 1 and 6)
  5. Trial for middle term (11x):
    • Correct combination: 3x * 3 + 2x = 11x
    • Result: (3x + 2)(x + 3)

Problem 2: Factor 2x² - 3x - 20

  1. List factors: 2x²
    • Factors: 2, 1
  2. List factors: -20
    • Factors: 1, 20; 2, 10; 4, 5
  3. Set up binomials and avoid GCF complications:
    • (2x - 5)(x + 4)
  4. Check middle term (-3x):
    • Result: Combines correctly

Problem 3: Factor 6x² + 25x - 9

  1. List factors: 6x²
    • Factors: 6, 1; 3, 2
  2. List factors: -9
    • Factors: 9, 1; 3, 3
  3. Set up binomials and check combinations:
    • Attempts and corrections
    • Final combination: (3x - 1)(2x + 3)

Problem 4: Factor 12x² + 61x + 70

  1. List factors: 12x²
    • Factors: 12, 1; 3, 4
  2. List factors: 70
    • Factors: 1, 70; 7, 10
  3. Attempts and checks:
    • Final result: (3x + 7)(4x + 10)

Problem 5: Factor 10x² - 68x + 48

  1. GCF: 2
  2. Reduced equation: 5x² - 34x + 24
  3. List factors and set up binomials:
    • Factors: 5, 1; 24 factors: 2, 12; 4, 6
  4. Final combination: (5x - 4)(x - 6)
  5. Include GCF:
    • Result: 2(5x - 4)(x - 6)

Conclusion

  • Recap: The guess and check method involves listing factors and strategically testing combinations to achieve the desired middle term.
  • Tip: Always check for a Greatest Common Factor (GCF) first.
  • Encouragement: Practice to improve efficiency.

End of Lecture

  • Closing: Thank you and hope the explanation was helpful. Bye-bye.

Full transcript