Overview
This lesson explains the zero product property and demonstrates how to use it to solve quadratic equations by setting each factor equal to zero.
Zero Product Property
- The zero product property states that if a × b = 0, then a = 0 or b = 0 (or both).
- If the product of two numbers is zero, at least one of the numbers must be zero.
- Multiplying zero by any number always results in zero.
Applying the Zero Product Property to Equations
- When solving quadratic equations in factored form, set each factor equal to zero.
- Example: For (x - 3)(x + 2) = 0, set x - 3 = 0 and x + 2 = 0, and solve each equation.
- Solving these gives the solutions x = 3 and x = -2.
Practice Problems and Solutions
- Problem 1: 3x(x - 7) = 0
- Set 3x = 0 ⇒ x = 0
- Set x - 7 = 0 ⇒ x = 7
- Problem 2: (2x - 3)(3x - 5) = 0
- Set 2x - 3 = 0 ⇒ x = 3/2
- Set 3x - 5 = 0 ⇒ x = 5/3
- Plugging the solutions back into the original equations confirms that the entire expression equals zero.
Steps to Solve Using Zero Product Property
- Ensure the equation is in factored form.
- Set each factor equal to zero.
- Solve each resulting simple equation for x.
Key Terms & Definitions
- Zero Product Property — If the product of factors equals zero, at least one factor must be zero.
- Factored Form — An expression written as a product of its factors, e.g., (x - 3)(x + 2).
Action Items / Next Steps
- Practice solving factored quadratic equations by applying the zero product property.
- Make sure to write equations in factored form before using this method.