Lecture Notes: Quadratic Equations

1. Recognizing Quadratic Relationships

• Definition: A quadratic relationship can be identified in three ways: equation, table, and graph.
  • Equation: For example, the equation $y = x^2 - 6x + 8$ represents a quadratic relationship because the highest exponent is 2.
  • Table: Analyze finite differences of Y values; constant second differences indicate a quadratic relationship.
  • Graph: The graph of a quadratic relationship is a U-shaped curve known as a parabola. Key features:
    • Vertex: The highest or lowest point.
    • Axis of Symmetry: Vertical line through the vertex.

2. Standard Form Equation

• Format: $y = ax^2 + bx + c$, where a, b, and c are real numbers.
  • a-Value: Indicates opening direction.
    • $a > 0$: Opens upwards.
    • $a < 0$: Opens downwards.
  • c-Value: Y-intercept at $(0, c)$.

3. Vertex Form Basics

• Format: $y = a(x - h)^2 + k$
  • Vertex: $(h, k)$
  • Axis of Symmetry: Line $x = h$
  • a-Value: Same directional indication as standard form.
  • Conversion: Convert from standard to vertex form via completing the square.

4. Factored Form Basics

• Format: $y = a(x - m)(x - n)$
  • X-Intercepts: Found at $m$ and $n$.
  • Axis of Symmetry: $x = \frac{m + n}{2}$
  • a-Value: Direction of opening.

5. Factoring Quadratics

• Purpose: Convert standard form to factored form.
• Methods:
  • Leading Coefficient of 1: Use product and sum method.
  • Leading Coefficient ≠ 1: Use splitting the middle term, factor by grouping.
  • Difference of Squares: Apply the rule $(a^2 - b^2) = (a - b)(a + b)$.

6. Solving Quadratic Equations by Factoring

• Steps:
  • Set equation to zero.
  • Factor the quadratic.
  • Use zero-product property to solve for x.

7. Solving by Completing the Square

• When to Use: When factoring is not feasible.
• Steps:
  • Rearrange to form a perfect square trinomial.
  • Factor and solve.

8. Solving Using the Quadratic Formula

• Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Usefulness: Solves any quadratic equation.
• Example: Plug coefficients into the formula to find solutions.

9. Understanding the Discriminant

• Definition: Part of the quadratic formula under the square root ($b^2 - 4ac$).
  • Positive: Two real solutions.
    • Perfect square: rational solutions.
    • Not a perfect square: irrational solutions.
  • Zero: One real solution.
  • Negative: No real solutions (complex solutions).

10. Finding the Vertex of a Parabola

• Methods:
  • Completing the Square: Convert to vertex form.
  • Average X-Intercepts: Useful after factoring.
  • Using Vertex Formula: $x = \frac{-b}{2a}$, then substitute into equation to find y.

Summary

These 10 must-know points about quadratic equations provide a comprehensive foundation for understanding and working with quadratic relationships in various forms. Let me know if you want further elaboration or a top 10 list on another topic!