quadratic equation in

The quadratic formula


for solving the quadratic equation

ax2+bx+c= 0 (1)

with real coefficients a, b, c is valid as well for all complex values of these coefficients (a0), when the square root is determined as is presented in the parent entry (http://planetmath.org/TakingSquareRootAlgebraically).

Proof.  Multiplying (1) by 4a and adding b2 to both sides gives an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) equation




or furthermore


Taking square root algebraically yields


which implies the quadratic formula.

Note.  A quadratic formula is meaningful besides also in other fields with characteristicPlanetmathPlanetmathPlanetmath 2  if one can find the needed “square root” (this may require a field extension).

Title quadratic equation in
Canonical name QuadraticEquationInmathbbC
Date of creation 2013-03-22 17:36:36
Last modified on 2013-03-22 17:36:36
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 30-00
Classification msc 12D99
Synonym quadratic equation
Related topic QuadraticFormula
Related topic DerivationOfQuadraticFormula
Related topic CardanosDerivationOfTheCubicFormula