quadratic equation in $ℂ$

The quadratic formula

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

for solving the quadratic equation

$a{x}^2+bx+c=\mathrm{\hspace{0.33em}0}$(1)

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

Proof.  Multiplying (1) by $4a$ and adding $b^2$ to both sides gives an equivalent (http://planetmath.org/Equivalent3) equation

$$4a^2{x}^2+4abx+4ac+b^2=b^2$$

or

$${(2ax)}^2+2\cdot 2ax\cdot b+b^2=b^2-4ac$$

or furthermore

$${(2ax+b)}^2=b^2-4ac.$$

Taking square root algebraically yields

$$2ax+b=\pm \sqrt{b^2-4ac},$$

which implies the quadratic formula.

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