Editing Quadratic equation (section)

===Generalization of quadratic equation===
The formula and its derivation remain correct if the coefficients {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are [[complex number]]s, or more generally members of any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is not {{math|2}}. (In a field of characteristic 2, the element {{math|2''a''}} is zero and it is impossible to divide by it.)

The symbol
<math display="block">\pm \sqrt {b^2-4ac}</math>
in the formula should be understood as "either of the two elements whose square is {{math|''b''<sup>2</sup> &minus; 4''ac''}}, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic {{math|2}}. Even if a field does not contain a square root of some number, there is always a quadratic [[extension field]] which does, so the quadratic formula will always make sense as a formula in that extension field.

====Characteristic 2====
In a field of characteristic {{math|2}}, the quadratic formula, which relies on {{math|2}} being a [[unit (ring theory)|unit]], does not hold. Consider the [[monic polynomial|monic]] quadratic polynomial
<math display="block">x^{2} + bx + c</math>
over a field of characteristic {{math|2}}. If {{math|''b'' {{=}} 0}}, then the solution reduces to extracting a square root, so the solution is
<math display="block">x = \sqrt{c}</math>
and there is only one root since
<math display="block">-\sqrt{c} = -\sqrt{c} + 2\sqrt{c} = \sqrt{c}.</math>
In summary,
<math display="block">\displaystyle x^{2} + c = (x + \sqrt{c})^{2}.</math>
See [[quadratic residue]] for more information about extracting square roots in finite fields.

In the case that {{math|''b'' &ne; 0}}, there are two distinct roots, but if the polynomial is [[irreducible polynomial|irreducible]], they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the '''2-root''' {{math|''R''(''c'')}} of {{math|''c''}} to be a root of the polynomial {{math|''x''<sup>2</sup> + ''x'' + ''c''}}, an element of the [[splitting field]] of that polynomial. One verifies that {{math|''R''(''c'') + 1}} is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}} are
<math display="block">\frac{b}{a}R\left(\frac{ac}{b^2}\right)</math>
and
<math display="block">\frac{b}{a}\left(R\left(\frac{ac}{b^2}\right)+1\right).</math>

For example, let {{math|''a''}} denote a multiplicative generator of the group of units of {{math|''F''<sub>4</sub>}}, the [[Galois field]] of order four (thus {{math|''a''}} and {{math|''a'' + 1}} are roots of {{math|''x''<sup>2</sup> + ''x'' + 1}} over {{math|''F''<sub>4</sub>}}. Because {{math|(''a'' + 1)<sup>2</sup> {{=}} ''a''}}, {{math|''a'' + 1}} is the unique solution of the quadratic equation {{math|''x''<sup>2</sup> + ''a'' {{=}} 0}}. On the other hand, the polynomial {{math|''x''<sup>2</sup> + ''ax'' + 1}} is irreducible over {{math|''F''<sub>4</sub>}}, but it splits over {{math|''F''<sub>16</sub>}}, where it has the two roots {{math|''ab''}} and {{math|''ab'' + ''a''}}, where {{math|''b''}} is a root of {{math|''x''<sup>2</sup> + ''x'' + ''a''}} in {{math|''F''<sub>16</sub>}}.

This is a special case of [[Artin–Schreier theory]].