Notes on Quadratic Equations

Introduction to Quadratic Equations

• Study the chapter on Quantic Equations carefully.
• Begin with the quadratic expression.

Quadratic Expression

• Form: ax² + bx + c
• Called quadratic because the highest degree is 2.
• Represents a relationship between two variables: x and y.
• Can be plotted as a graph: Parabola.

Graph of Quadratic Expression

• The graph of a quadratic expression is a parabola.
• Parabolas can open:
  • Upwards (mouth of parabola opens up)
  • Downwards (mouth of parabola opens down)
• Important Points:
  • Vertex: Point where the parabola turns.
  • Legs of the parabola go to infinity:
    • Upwards: legs go to +∞.
    • Downwards: legs go to -∞.

Determining the Direction of Opening

• To determine if the parabola opens up or down:
  • Look at the coefficient of x² (denoted as a).
    • If a > 0, parabola opens upwards.
    • If a < 0, parabola opens downwards.

Interaction with the X-axis

• To determine if the parabola touches, cuts, or is away from the x-axis, check the discriminant:
  • Discriminant = b² - 4ac
  • If D < 0: Parabola is away from the x-axis.
  • If D = 0: Parabola touches the x-axis (1 point).
  • If D > 0: Parabola cuts the x-axis (2 points).

Vertex Coordinates

• The x-coordinate of the vertex is given by:
  • x = -b / 2a
• The y-coordinate of the vertex is given by:
  • y = -D / 4a

Minimum and Maximum Values

• If the parabola opens upwards:
  • Minimum value of y is -D/4a.
  • Maximum value of y is +∞.
• If the parabola opens downwards:
  • Minimum value of y is -∞.
  • Maximum value of y is -D/4a.

Conditions for Quadratic Expressions

• Always positive:
  • If a > 0 and D < 0.
• Always negative:
  • If a < 0 and D < 0.
• Greater than or equal to 0:
  • If a > 0 and D ≤ 0.
• Less than or equal to 0:
  • If a < 0 and D ≤ 0.

Problem-solving Approach

• Example Problem: If a quadratic expression has no real roots, it means it does not intersect the x-axis. This implies it remains either always positive or always negative.
• To show that C(a+b+c) > 0 when there are no real roots:
  • Evaluate function values at specific points.
  • Both values must share the same sign for the product to be positive.

Additional Example Problems

1. Finding values so that a quadratic expression is always positive: Analyze conditions on a and D to ensure positivity.
2. Finding values so that a quadratic expression is always negative: Similarly, analyze conditions.

Conclusion

• Understanding the properties of quadratic expressions is crucial in solving related problems.
• Graph analysis helps in determining the nature of the quadratic expression.