Quadratic Equations: Discriminant and Roots

• These are study notes on the theory of quadratic equations
• These notes cover the discriminant, nature of roots, and solving for roots

Quadratic Equations

• A quadratic equation is expressed as: 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0
• The quadratic formula provides the solutions for x: 𝑥 = (−𝑏 ± √(𝑏² − 4𝑎𝑐)) / 2𝑎

Discriminant

• The discriminant of a quadratic equation is the expression 𝑏² − 4𝑎𝑐
• It determines the nature of the roots

Nature of Roots

• Case 1: If 𝑏² − 4𝑎𝑐 = 0, the roots are real, equal, and rational
• Example x² - 8x + 16 = 0 roots are real, equal, and rational
• Case 2: If 𝑏² − 4𝑎𝑐 < 0, the roots are unequal and imaginary
  • Example 4𝑥² + 𝑥 + 1 = 0 roots are unequal and imaginary
• Case 3: If 𝑏² − 4𝑎𝑐 > 0, then:
  • If 𝑏² − 4𝑎𝑐 is a perfect square, the roots are real, unequal, and rational
  • If 𝑏² − 4𝑎𝑐 is not a perfect square, the roots are real, unequal, and irrational
• Example: 6𝑥² − 𝑥 − 15 = 0 , roots are real, unequal and rational

Discriminant Examples

• For x² + 9x + 2 = 0, the discriminant is 73, and roots are real, unequal, and irrational
• For 4x² – 5x + 1 = 0, the discriminant is 9
• The equation x² - 4x + 13 = 0 has a discriminant of -36

Determining the set of values Example 1

• Finding values of k for real roots
• For 𝑘𝑥² + 4𝑥 + 1 = 0, the discriminant condition for real roots becomes 16 − 4𝑘 ≥ 0, leading to 𝑘 ≤ 4
• For 2𝑥² + 𝑘𝑥 + 3 = 0, the discriminant condition leads to 𝑘 ≥ 2√6 or 𝑘 ≤ −2√6

Verifying Nature of Factors

• The nature of roots can be verified by factorization
• x² - 6x + 9 is factored to show real and equal roots and the result being proven
• x² + 5x + 6 uses factorization to get real, unequal, and rational roots with result verified

Determining Roots of Quadratic Equations

• It is possible to determine the nature of the roots of quadratic equations without solving them
• Example 3𝑥² − 4𝑥 + 6 = 0 has unequal and imaginary roots
• Example 2𝑥² − 7𝑥 = −1 gives real, unequal, and irrational roots

Cube Root of Unity

• Let x be the cube root of 1
• 𝑥 = 1, (−1 + 𝑖√3)/2 AND (−1 − 𝑖√3)/2
• Where (−1 + 𝑖√3)/2 is ω and (−1 − 𝑖√3)/2 is its compliment

Examples of Cube Root of Unity

• All cases can be expressed as 1, ω and ω²
• The sum of cube roots of unity is zero ie: 1 + ω + ω² = 0
• The results of reciprocals of cube roots is as follows: ω = 1/(ω²), and (ω²) = 1/ω

Additional Theorems

• Further properties are presented
• For example 1+ ω =−ω²
• Show that: x³ + y³ = (x + y)(x – wy)(x – w²y)

Evaluating roots

• The values are determined that satisfy the following: 15 = 1, 24 = 1, and 90 = 1
• If ω = (−1 + 𝑖√3)/2 and w² = (−1 − 𝑖√3)/2
• For example (-1 + i√3) + (-1 + I√3) = 16

Working with Real Numbers

• Example - Find the cube roots of -1
• 𝑥 = −1, 𝑥 = (−1 ± 𝑖√3)/2
• Which is expressed as, −1,−𝜔,−𝜔2

Roots for Quadratics

• Given information to solve for: x² + 3x + k = 0, gives the value of k with certain conditions
  • If values are real, discriminant: 𝑏² − 4𝑎𝑐 ≥ 0, then 𝑘 ≤ 9/8
  • Imaginary: 𝑘 > 9/8
• If information to solve for: x² − 5x + k = 0, and given that values differ by unity.