Overview
This lecture covers methods for solving quadratic equations, including graphing, factoring, and using square roots, and provides several worked examples and applications in real-life scenarios.
Understanding Quadratic Equations
- A quadratic equation is an equation of the form ax² + bx + c = 0.
- Solutions to quadratics (roots) represent x-values where the graph crosses the x-axis (x-intercepts).
- Setting y (or f(x)) to zero allows solving for these roots.
Methods for Solving Quadratic Equations
- Factoring: Rewrite the equation as a product of binomials and set each factor equal to zero.
- Square Roots: Isolate the squared term and take the square root of both sides, remembering to use ± for both positive and negative roots.
- Graphing: Plot the function and identify the x-intercepts visually.
- Zero Product Property: If ab = 0, then a = 0 or b = 0.
Solving by Graphing Examples
- Set quadratic equal to zero and plot to find x-intercepts, e.g., axis of symmetry at x = -b/2a.
- Recognize no solution exists if the graph does not cross the x-axis.
Solving by Square Roots Examples
- Get variable squared by itself, then take square roots of both sides.
- Rationalize denominators if necessary.
- Only positive times are valid in real-world (time) scenarios.
Solving by Factoring Examples
- When the quadratic factors easily, write as (x - r₁)(x - r₂) = 0.
- If difference of squares: a² - b² = (a + b)(a - b).
- Set each factor to zero to get solutions.
Word Problem Application: Maximizing Revenue
- Revenue function can be modeled as a quadratic with intercepts at key price points.
- Maximum revenue occurs at the vertex, halfway between x-intercepts.
- Use context to set up quadratic and solve for maximum.
Physics Application: Object Drop Problem
- Use h = -16t² + h₀ to model dropped objects.
- Set height to zero and solve for time with square roots.
- Only consider positive solution for time.
Key Terms & Definitions
- Quadratic Equation — An equation where the highest power of x is 2.
- Root/Solution — x-value(s) where the quadratic equals zero.
- Factoring — Expressing a quadratic as a product of binomials.
- Zero Product Property — If a product is zero, at least one factor must be zero.
- Axis of Symmetry — x = -b/2a; vertical line passing through the vertex of the parabola.
- Vertex — The maximum or minimum point of a parabola.
- Difference of Squares — A factoring pattern: a² - b² = (a + b)(a - b).
Action Items / Next Steps
- Practice solving quadratics by factoring, square roots, and graphing.
- Complete assigned textbook problems on section 3.1 methods.
- Review key terms and ensure understanding of each solution method.