Discriminant

In algebra, the discriminant, sometimes represented by the symbol [math]\displaystyle{ \Delta }[/math], is an algebraic expression used to determine the number of roots a polynomial have.

For example, the discriminant of the quadratic polynomial

[math]\displaystyle{ ax^2+bx+c\, }[/math]       is       [math]\displaystyle{ \,b^2-4ac }[/math].

If the discriminant is larger than zero, then the polynomial has two (distinct) real numbers as roots.

If the discriminant is equal to zero, then the polynomial has two repeating real numbers as roots (i.e., exactly one real root).

If the discriminant is smaller than zero, then the polynomial has two (purely) complex numbers as roots (i.e., zero real root).

Apart for quadratic polynomials, discriminants can be defined for cubic polynomials, general conic equations, and other mathematical entities such as differential equations and quadratic forms as well.

Discriminant Media

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  The zero set of discriminant of the cubic x3 + bx2 + cx + d, i.e. points satisfying b2c2 – 4c3 – 4b3d – 27d2 + 18bcd = 0.

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  The discriminant of the quartic polynomial x4 + cx2 + dx + e. The surface represents points (c, d, e) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.

Related pages

• Equation solving
• Quadratic equation