Solving Quadratic Equations by Factoring

Overview

• Today’s topic: Solving quadratic equations by factoring.
• Method overview: One way to find the solutions of quadratic equations.

Example Problems

Problem 1: 3x² - 9x = 0

1. Identify common monomial factor (CMF): 3x.
2. Factor the equation: 3x(x - 3) = 0.
3. Set each factor to zero:
   • 3x = 0 → x = 0
   • x - 3 = 0 → x = 3
4. Solutions: x = 0, x = 3.

Problem 2: 5x² - 20x = 0

1. CMF is 5x.
2. Factor the equation: 5x(x - 4) = 0.
3. Set each factor to zero:
   • 5x = 0 → x = 0
   • x - 4 = 0 → x = 4
4. Solutions: x = 0, x = 4.

Example Problems with Difference of Squares

Problem 3: x² - 16 = 0

1. Recognize as difference of squares: a² - b² = (a + b)(a - b).
2. Identify: a = x, b = 4.
3. Factor: (x + 4)(x - 4) = 0.
4. Set each factor to zero:
   • x + 4 = 0 → x = -4
   • x - 4 = 0 → x = 4
5. Solutions: x = -4, x = 4.

Problem 4: x² - 64 = 0

1. Identify: a = x, b = 8.
2. Factor: (x + 8)(x - 8) = 0.
3. Set each factor to zero:
   • x + 8 = 0 → x = -8
   • x - 8 = 0 → x = 8
4. Solutions: x = -8, x = 8.

Trinomial Quadratic Equations

Problem 5: x² - 8x + 15 = 0

1. Identify factors of 15 that sum to -8: -5 and -3.
2. Factor: (x - 5)(x - 3) = 0.
3. Set each factor to zero:
   • x - 5 = 0 → x = 5
   • x - 3 = 0 → x = 3
4. Solutions: x = 5, x = 3.

Problem 6: x² - 7x + 12 = 0

1. Identify factors of 12 that sum to -7: -4 and -3.
2. Factor: (x - 4)(x - 3) = 0.
3. Set each factor to zero:
   • x - 4 = 0 → x = 4
   • x - 3 = 0 → x = 3
4. Solutions: x = 4, x = 3.

Homework Problems

1. Solve: x² - 100 = 0
2. Solve: x² + 5x + 6 = 0

Conclusion

• Reminder for viewers: Like, subscribe, and hit the bell button for updates on new content.