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Generalization of quadratic equation

The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2 a is zero and it is impossible to divide by it.)

The symbol ± b 2 − 4 a c {\displaystyle \pm {\sqrt {b^{2}-4ac}}} in the formula should be understood as "either of the two elements whose square is b − 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 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.