Quick Recap — Quadratic Equations in Depth

  • Symmetric functions: α2+β2=(α+β)2−2αβ\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta, α3+β3=(α+β)3−3αβ(α+β)\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta), ∣α−β∣=D/∣a∣|\alpha-\beta|=\sqrt{D}/|a|.
  • Nature/condition: equal roots ⇒D=0\Rightarrow D=0; real roots ⇒D≥0\Rightarrow D\ge0.
  • Transformed equations: roots reciprocal ⇒\Rightarrow reverse coefficients; complex roots occur in conjugate pairs.
  • Range of a quadratic by completing the square; extremum at x=−b2ax=-\tfrac{b}{2a}.