Overview

This topic focuses on quadratic equations and inequalities, exploring how to solve, factor, graph, and analyze parabolas. You'll learn how to complete the square, use the quadratic formula, and determine the nature of roots based on the discriminant. These concepts are foundational for understanding quadratic functions and their real-world applications.

Key Concepts and Structures

• Standard Form: A quadratic equation is written as ax^2 + bx + c = 0, where a ≠ 0.
• Factoring: When possible, express the quadratic as (x - r)(x - s) = 0 and solve for roots.
• Completing the Square: Rearranging to make a perfect square trinomial. Used when factoring is not straightforward.
• Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a). Works for all quadratics.
• Discriminant: Part of the quadratic formula: D = b² - 4ac
  • D > 0: Two real roots
  • D = 0: One real root
  • D < 0: Two complex roots
• Vertex Form: y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
• Graphing: Identify direction (opens up or down), axis of symmetry (x = -b/2a), and vertex.
• Quadratic Inequalities: Solve ax^2 + bx + c < 0 or > 0 by finding x-intercepts and analyzing intervals.

Step-by-Step Example

Problem: Solve the equation 2x^2 - 4x - 6 = 0 using the quadratic formula.

Step 1: Identify a = 2, b = -4, c = -6

Step 2: Plug into the formula:
x = [-(-4) ± √((-4)^2 - 4(2)(-6))] / (2 * 2)
x = [4 ± √(16 + 48)] / 4 = [4 ± √64] / 4

Step 3: Simplify:
x = [4 ± 8] / 4

Step 4: Find both roots:
x = (4 + 8)/4 = 3, x = (4 - 8)/4 = -1

Final Answer: x = 3 and x = -1

Quick Tip