Overview
This lecture explains how to find the zeros of a quadratic equation when the leading coefficient is not one, using factoring and the zero-product property.
Identifying Coefficients
- For the equation 3x² + 7x + 2 = 0: a = 3, b = 7, c = 2.
- Identify a (coefficient of x²) and c (constant term) first.
Factoring Using the AC Method
- Multiply a * c: 3 * 2 = 6.
- Find two factors of 6 that add up to b (7); factors are 6 and 1.
- Rewrite the middle term (7x) as 6x + 1x to split the equation: 3x² + 6x + 1x + 2.
- Factor by grouping: (3x² + 6x) + (1x + 2) → 3x(x + 2) + 1(x + 2).
- Factor common binomial: (x + 2)(3x + 1).
Finding the Zeros
- Set each factor equal to zero: x + 2 = 0 and 3x + 1 = 0.
- Solve for x: x = -2 and x = -1/3.
Key Terms & Definitions
- Zero of an Equation — A value of x that makes the equation equal zero.
- Factoring — Rewriting an equation as a product of simpler expressions.
- Zero Product Property — If ab = 0, then a = 0 or b = 0.
- AC Method — Multiply a and c, find factor pairs that sum to b.
Action Items / Next Steps
- Practice factoring quadratics where a > 1 using the AC method.
- Solve for zeros after factoring by setting each factor to zero.