Lecture on Quadratics

Introduction

• Start of Chapter 2: Quadratics
• Skills to master:
  • Solving quadratic equations
  • Completing the square
  • Quadratic functions
  • Quadratic graphs
  • The discriminant (new topic)
• Application of Quadratics:
  • Projectile motion
  • Distance and displacement forecasting

Solving Quadratic Equations

By Factorization

1. Ensure the equation equals zero.
2. Identify two numbers that multiply to give the constant term and add to give the coefficient of the middle term.
3. Factor form:
   • Example: x^2 + 5x – 6 = 0 ⟶ (x – 1)(x + 6) = 0
4. Solving for x:
   • x = -6 or x = 1

Using the Quadratic Formula

1. For an equation ax^2 + bx + c = 0
2. Quadratic formula:
   • x = [-b ± sqrt(b^2 – 4ac)] / (2a)
3. Example:
   • a = 1, b = 5, c = -6
   • x = [-5 ± sqrt(5^2 – 4(1)(-6))] / (2 * 1) = x = -6 or x = 1

Roots and Discriminant

• Ensure you understand both factorization and quadratic formula methods well as they will be crucial for working with the discriminant.
• Discriminant Overview:
  • Part of the quadratic formula under the square root: b^2 – 4ac
  • Determines the nature of the roots:
    • > 0: Two real roots (crosses x-axis twice)
    • = 0: One real/equal root (touches x-axis once)
    • < 0: No real roots (does not cross x-axis)

Quadratics in Disguise

• Sometimes equations do not appear as straightforward quadratics (like x terms in terms of halves).
• Approach:
  1. Rewrite in quadratic form.
  2. Use substitution (e.g., let y = x^(1/2)) to solve.
  3. Restore original variables to get final answer.
• Example:
  • x^(1/2)^2 - 6x^(1/2) + 8 = 0
  • Let y = x^(1/2):
  • (y - 2)(y - 4) = 0 ⟶ y = 2 or 4
  • x = 4 and x = 16

Completing the Square

• Convert quadratic expression to (x ± p)^2 - q form to find roots or sketch graphs easily.
• Example:
  • x^2 + 12x:
  • (x + 6)^2 - 36
  • For solving: rearrange
    • (x + 6)^2 = k
    • x + 6 = ± sqrt(k)

Practice Problems

• Several exercises were introduced:
  1. Solve x^2 - 7x + 10 = 0 by factorization.
  2. Solve by using quadratic formula where necessary.
  3. Problems involve different cases (integer values, substitution method, etc.).

Functions and Graphs

• Function notation: f(x), g(x) etc.
• Domain: set of possible inputs (x-values).
• Range: set of possible outputs (y-values).
• One-to-one functions: each input has a unique output.
• Example: f(x) = x^2 - 3x
  • f(-4) = 28
  • Roots: x^2 - 3x = 0 (factorize for roots).

Exam Style Questions

Discriminant Problems

• Determine nature of roots using b^2 - 4ac directly.
  • Example:
    • For x^2 + 4x + 4 = 0:
    • b^2 - 4ac = 16 - 16 = 0 ⟶ one root.
• Equality and inequalities problems: Determine values that satisfy given conditions on roots.

Solving Quadratic Equations with Parameters

• E.g., The equation x^2 + 2px + 3p + 4 = 0 has equal roots:
  • Set up b^2 - 4ac = 0 and solve for p.

Homework and Exercises

• Practice complete multifaceted questions (e.g. solving by both methods).
• Additional assigned homework for Chapter 1 as preparatory work.
• Due: Sunday.

Key Takeaways

• Thorough understanding of solving quadratics by both factorization and formula is critical.
• Mastery of completing the square and function behavior analysis is essential.
• New topics: The discriminant provides deep insight into the nature of quadratic equation roots.
• Practical applications span several mathematical and real-world contexts.