Solving Quadratic Equations by Factoring

Jul 23, 2024

Solving Quadratic Equations by Factoring

Overview

  • Learn how to solve quadratic equations by factoring
  • Tutorial divided into two parts
    • Part 1: Two-term quadratics (using Greatest Common Factor, Difference of Squares)
    • Part 2: Trinomials (leading coefficient of one or different from one)
    • Also covers non-standard form and common mistakes

Part 1: Solving Two-Term Quadratic Equations

Greatest Common Factor (GCF)

  • Example 1:
    • Equation: x^2 - 10x = 0
    • GCF of x^2 and -10x is x
    • Factored form: x(x - 10)
    • Zero Product Property: x = 0 or x - 10 = 0
    • Solutions: x = 0 or x = 10
  • Example 2:
    • Equation: 3x^2 + 15x = 0
    • GCF of 3x^2 and 15x is 3x
    • Factored form: 3x(x + 5)
    • Solutions: x = 0 or x = -5

Difference of Two Squares

  • Example 3:
    • Equation: x^2 - 81 = 0
    • 81 is a perfect square: 9^2
    • Factored form: (x + 9)(x - 9)
    • Solutions: x = -9 or x = 9
  • Example 4:
    • Equation with negative leading coefficient: -x^2 + 49 = 0
    • Factor out -1: -(x^2 - 49) = 0
    • Factored form: - (x + 7)(x - 7)
    • Solutions: x = -7 or x = 7

Part 2: Solving Quadratic Trinomials

Leading Coefficient of One

  • Example 5: Product-Sum Method
    • Equation: x^2 + 11x + 24 = 0
    • Product of 24, sum of 11: Factor pairs (3, 8)
    • Factored form: (x + 3)(x + 8)
    • Solutions: x = -3 or x = 8
  • Example 6: Negative Product
    • Equation: x^2 - 5x - 36 = 0
    • Product of -36, sum of -5: Factor pairs (4, -9)
    • Factored form: (x + 4)(x - 9)
    • Solutions: x = -4 or x = 9

Leading Coefficient Not Equal to One

  • Example 7:
    • Equation: -x^2 + 42x + 63 = 0
    • Factor out -1: -(x^2 - 42x - 63) = 0
    • Product-Sum Method: Factors (3, 14)
    • Factored form: -(x - 3)(x + 14)
    • Solutions: x = 3 or x = -14
  • Example 8: AC Method
    • Equation: 5x^2 + 18x + 9 = 0
    • Multiply leading coefficient by constant: 5 * 9 = 45
    • Factors of 45 that add to 18: (3, 15)
    • Split middle term, factor by grouping:
      • Group 1: 5x^2 + 15x (GCF = 5x)
      • Group 2: 3x + 9 (GCF = 3)
      • Factored form: (5x + 3)(x + 3)
    • Solutions: x = -3/5 or x = -3

Non-Standard Form

  • Example 9:
    • Initial Equation: 4x(x - 5) = -25
    • Distribute and rearrange: 4x^2 - 20x + 25 = 0
    • Use AC Method:
      • Product is 100, sum is -20
      • Factors: (-10, -10)
    • Factored form: (2x - 5)^2 = 0
    • Solution: x = 5/2