Quadratic equation

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An algebraic equation of the second degree. The general form of a quadratic equation is

In the field of complex numbers a quadratic equation has two solutions, expressed by radicals in the coefficients of the equation:


When both solutions are real and distinct, when , they are complex (complex-conjugate) numbers, when the equation has the double root .

For the reduced quadratic equation

formula (*) has the form

The roots and coefficients of a quadratic equation are related by (cf. Viète theorem):


The expression is called the discriminant of the equation. It is easily proved that , in accordance with the fact mentioned above that the equation has a double root if and only if . See also Discriminant. Formula (*) holds also if the coefficients belong to a field with characteristic different from 2.

Formula (*) follows from writing the left-hand side of the equation as (splitting of the square).


[a1] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. Sect. 1.20
How to Cite This Entry:
Quadratic equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article