quadratic equation in ℂ (original) (raw)

The quadratic formula

for solving the quadratic equation

with real coefficients a, b, c is valid as well for all complex values of these coefficients (a≠0), when the square root is determined as is presented in the parent entry (http://planetmath.org/TakingSquareRootAlgebraically).

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

4⁢a2⁢x2+4⁢a⁢b⁢x+4⁢a⁢c+b2=b2

or

(2⁢a⁢x)2+2⋅2⁢a⁢x⋅b+b2=b2-4⁢a⁢c

or furthermore

Taking square root algebraically yields

which implies the quadratic formula.

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