Overview
This lecture explains how to solve quadratic equations using the quadratic formula, including identifying coefficients and simplifying solutions.
Quadratic Formula Basics
- The quadratic formula solves equations of the form ax² + bx + c = 0.
- The formula is: x = [-b ± √(b² - 4ac)] / (2a).
- Coefficients: a is in front of x², b in front of x, c is the constant term.
Example 1: Solving 2x² + 3x - 2 = 0
- Identify a = 2, b = 3, c = -2.
- Substitute into the formula: x = [-3 ± √(9 + 16)] / 4.
- Simplify: x = [-3 ± 5] / 4, giving x = (2/4) and x = (-8/4).
- Reduce: x = 1/2 and x = -2.
- Verification: Plugging x = -2 into the equation results in 0, so it is a valid solution.
Example 2: Solving 6x² - 17x + 12 = 0
- Identify a = 6, b = -17, c = 12.
- Substitute into the formula: x = [17 ± √(289 - 288)] / 12.
- Simplify: x = [17 ± 1] / 12, resulting in x = (18/12) and x = (16/12).
- Reduce: x = 3/2 and x = 4/3.
Key Terms & Definitions
- Quadratic Equation — An equation in the form ax² + bx + c = 0.
- Quadratic Formula — The formula x = [-b ± √(b² - 4ac)] / (2a) for solving quadratic equations.
- Coefficients — a, b, and c: numerical factors in the quadratic equation.
Action Items / Next Steps
- Practice using the quadratic formula on other equations.
- Double-check answers by substituting solutions back into original equations.