discriminant - Discriminant of a polynomial, Discriminant of a conic section, Discriminant of a quadratic form

A mathematical expression which shows whether a quadratic equation has real distinct roots, equal roots, or no real roots. The discriminant of the quadratic equation ax² + bx + c = 0 is b² ? 4ac. If b² ? 4ac > 0, the quadratic has real distinct roots; if b² ? 4ax = 0, it has equal roots; and if b² ? 4ac < 0, it has no real roots. The roots are then expressed in terms of complex numbers.

For a polynomial P(x) = a0 + a1x + a2x² + ... , the discriminant is a quantity D = D(a0,a1,a2,...) that equals 0 precisely for those P(x) that have at least one multiple root.

For a quadratic equation, the discriminant is the square-rooted section of the Quadratic Formula because you can use it to discriminate between whether the given quadratic has two solutions, one solution, or no solutions.

For instance, the quadratic polynomial P(x) = ax − 4ac, which is the quantity under the square root sign in the quadratic formula.

Discriminant of a polynomial

A discriminant of a polynomial is a number that can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root.

For the general definition, suppose

is a polynomial with real coefficients. The discriminant of this polynomial is defined as the determinant of the (2n − 1)×(2n − 1) matrix

In the case n = 4, this discriminant looks like this:

The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x), where p'(x) is the derivative of p(x).

One can show that, up to sign, the discriminant is equal to

where r1, ..., rn are the complex roots (counting multiplicity) of the polynomial p(x):

In fact, some authors define the discriminant by that formula, then show that the sign difference to the resultant is (−1) .

In order to compute discriminants, one does not evaluate the above determinant each time for different coefficients, but instead evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree p(x) = a3x + a1x + a0 is

The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above.

Discriminant of a conic section

For a conic section defined by the real polynomial:

ax + dx + ey + f= 0,

the discriminant is equal to

b2 − 4ac,

and determines the shape of the conic section.

Discriminant of a quadratic form

There is a substantive generalisation to quadratic forms Q over any field K of characteristic ≠ 2.

Discriminant of an algebraic number field

See main article, Discriminant of an algebraic number field.