quadratic equation in
for solving the quadratic equation
(1) |
with real coefficients , , is valid as well for all complex values of these coefficients (), when the square root is determined as is presented in the parent entry (http://planetmath.org/TakingSquareRootAlgebraically).
Proof. Multiplying (1) by and adding to both sides gives an equivalent (http://planetmath.org/Equivalent3) equation
or
or furthermore
Taking square root algebraically yields
which implies the quadratic formula.
Note. A quadratic formula is meaningful besides also in other fields with characteristic 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 |