Solving Quadratic Equations by Factoring

Jul 21, 2024

Solving Quadratic Equations by Factoring

Example 1: Difference of Perfect Squares

  • Equation: x^2 - 49 = 0
  • Steps:
    • Recognize it as a difference of perfect squares.
    • Factor: (x + 7)(x - 7)
    • Solve: Set each factor to zero
      • x + 7 = 0 → x = -7
      • x - 7 = 0 → x = 7

Example 2: Greatest Common Factor (GCF)

  • Equation: 3x^2 - 75 = 0
  • Steps:
    • Factor out the GCF (3)
      • 3(x^2 - 25) = 0
    • Factor the inner equation using difference of perfect squares
      • (x + 5)(x - 5)
    • Solve: Set each factor to zero
      • x + 5 = 0 → x = -5
      • x - 5 = 0 → x = 5

Example 3: Another Perfect Squares

  • Equation: 9x^2 - 64 = 0
  • Steps:
    • Recognize it as a difference of perfect squares
    • Factor: (3x + 8)(3x - 8)
    • Solve: Set each factor to zero
      • 3x + 8 = 0 → 3x = -8 → x = -8/3
      • 3x - 8 = 0 → 3x = 8 → x = 8/3

Example 4: Trinomial with Leading Coefficient of 1

  • Equation: x^2 - 2x - 15 = 0
  • Steps:
    • Find two numbers that multiply to -15 and add to -2
      • Numbers: -5 and 3
    • Factor: (x - 5)(x + 3)
    • Solve: Set each factor to zero
      • x - 5 = 0 → x = 5
      • x + 3 = 0 → x = -3

Example 5: Trinomial with Leading Coefficient of 1

  • Equation: x^2 + 3x - 28 = 0
  • Steps:
    • Find two numbers that multiply to -28 and add to 3
      • Numbers: -4 and 7
    • Factor: (x - 4)(x + 7)
    • Solve: Set each factor to zero
      • x - 4 = 0 → x = 4
      • x + 7 = 0 → x = -7

Example 6: Trinomial with Leading Coefficient not 1

  • Equation: 8x^2 + 2x - 15 = 0
  • Steps:
    • Multiply 8 and -15: -120
    • Find two numbers that multiply to -120 and add to 2
      • Numbers: 12 and -10
    • Replace 2x with 12x and -10x, factor by grouping
      • 8x^2 + 12x - 10x - 15
      • Group: 4x(2x + 3) - 5(2x + 3)
    • Final factor: (4x - 5)(2x + 3)
    • Solve: Set each factor to zero
      • 4x - 5 = 0 → x = 5/4
      • 2x + 3 = 0 → x = -3/2

Quadratic Formula

  • Standard Form: ax^2 + bx + c = 0
  • Formula: x = [-b ± √(b² - 4ac)] / 2a

Example: x^2 - 2x - 15

  • Identify: a = 1, b = -2, c = -15
  • Solution:
    • b² - 4ac = (-2)² - 4(1)(-15) = 4 + 60 = 64
    • √64 = 8
    • Solutions: x = [-(-2) ± 8] / 2
      • x = (2 + 8)/2 = 5
      • x = (2 - 8)/2 = -3

Example: 8x^2 + 2x - 15

  • Identify: a = 8, b = 2, c = -15
  • Solution:
    • b² - 4ac = 2² - 4(8)(-15) = 4 + 480 = 484
    • √484 = 22
    • Solutions: x = [-2 ± 22] / 16
      • x = (20/16) = 5/4
      • x = (-24/16) = -3/2

Conclusion: Quadratic equations can be solved by both factoring and using the quadratic formula.