Overview
This lecture explains how to solve quadratic equations using the quadratic formula with step-by-step examples.
Quadratic Formula Introduction
- The quadratic formula solves equations in the form ax² + bx + c = 0.
- The formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Identify a, b, and c from the equation: a is the coefficient of x², b is the coefficient of x, c is the constant.
Example 1: Solving 2x² + 3x - 2 = 0
- Assign: a = 2, b = 3, c = -2.
- Plug into formula: ( x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2 \times 2} ).
- Simplify: ( x = \frac{-3 \pm \sqrt{9 + 16}}{4} ) → ( x = \frac{-3 \pm 5}{4} ).
- Calculate both solutions: ( x = \frac{2}{4} = \frac{1}{2} ), ( x = \frac{-8}{4} = -2 ).
- Check answers by substituting back into the original equation.
Example 2: Solving 6x² - 17x + 12 = 0
- Assign: a = 6, b = -17, c = 12.
- Plug into formula: ( x = \frac{17 \pm \sqrt{(-17)^2 - 4(6)(12)}}{2 \times 6} ).
- Simplify: ( x = \frac{17 \pm \sqrt{289 - 288}}{12} ) → ( x = \frac{17 \pm 1}{12} ).
- Calculate both solutions: ( x = \frac{18}{12} = \frac{3}{2} ), ( x = \frac{16}{12} = \frac{4}{3} ).
Key Terms & Definitions
- Quadratic Equation — An equation of the form ax² + bx + c = 0.
- Quadratic Formula — A method for solving quadratic equations: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Discriminant — The expression ( b^2 - 4ac ) under the square root, determines the nature of the solutions.
Action Items / Next Steps
- Practice solving quadratic equations using the quadratic formula.
- Check solutions by substituting them back into the original equation.