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

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

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

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

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

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