Quadratic Equations & Expressions

Quadratic Equation

• General form: ax² + bx + c = 0, where a ≠ 0
• Coefficients: a, b, c
• Roots: α, β
• Sum of roots: α + β = −b/a
• Product of roots: αβ = c/a
• Discriminant: D = b² − 4ac
• Roots: x = (−b ± √D) / (2a)

Nature of Roots

• D > 0: Roots are real and distinct
• D = 0: Roots are real and equal
• D < 0: Roots are imaginary (complex)
• If D is a perfect square and a, b, c are rational ⇒ roots are rational
• If D > 0 and not a perfect square ⇒ roots are irrational

Formation of Quadratic Equation (Given Roots)

• If roots are α, β:
• x² − (α + β)x + αβ = 0

Transformation of Equations

• If α, β are roots of ax² + bx + c = 0:
• Roots 1/α, 1/β ⇒ cx² + bx + a = 0 (replace x by 1/x)
• Roots −α, −β ⇒ ax² − bx + c = 0 (replace x by −x)
• Roots α − k, β − k ⇒ replace x by (x − k)
• Roots kα, kβ ⇒ replace x by x/k
• Roots α/k, β/k ⇒ replace x by kx
• Roots αⁿ, βⁿ ⇒ replace x by x¹⁄ⁿ

Roots in Special Cases (Sign Analysis)

• If a > 0, b > 0, c > 0 ⇒ both roots negative
• If a > 0, b < 0, c > 0 ⇒ both roots positive
• If c < 0 ⇒ roots are of opposite signs
• Magnitude comparison depends on sign of b

Condition for Common Root(s)

• For equations ax² + bx + c = 0 and dx² + ex + f = 0:
• Condition for one common root:
• (dc − af)² = (bf − ce)(ae − bd)
• Condition for both roots common:
• a/d = b/e = c/f

Greatest & Least Value of Quadratic Expression

• Expression: ax² + bx + c
• If a > 0 ⇒ minimum value at x = −b/(2a)
• Minimum value: (4ac − b²) / (4a)
• If a < 0 ⇒ maximum value at x = −b/(2a)
• Maximum value: (4ac − b²) / (4a)

Location of Roots

• Let f(x) = ax² + bx + c
• A number k lies between roots if:
• f(k) < 0 and D > 0
• Roots are of opposite signs if f(0) < 0
• Both roots > k if:
  • D ≥ 0
  • −b/(2a) > k
  • f(k) > 0
• Both roots < k if:
  • D ≥ 0
  • −b/(2a) < k
  • f(k) > 0
• Roots lie in (k₁, k₂) if:
  • D ≥ 0
  • k₁ < −b/(2a) < k₂
  • f(k₁) > 0 and f(k₂) > 0
• Exactly one root in (k₁, k₂) if f(k₁) · f(k₂) < 0

Theory of Equations (Quick Facts)

• Polynomial of degree n has n roots (real or complex)
• Odd-degree polynomial has at least one real root
• If α is a root ⇒ (x − α) is a factor
• Complex and irrational roots occur in conjugate pairs
• Descartes’ Rule of Signs:
  • Max positive roots = number of sign changes in p(x)
  • Max negative roots = sign changes in p(−x)

Quadratic Expression in Two Variables

• General form: ax² + 2hxy + by² + 2gx + 2fy + c
• Condition to factor into linear factors:
• abc + 2fgh − af² − bg² − ch² = 0
• Or determinant form equals zero
• If one root is k times the other: (k + 1)² b² = 4kac