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where the discriminant is zero if and only if the two roots are equal. If a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two complex conjugate roots if it is negative. [6] The discriminant is the product of a 2 and the square of the difference of the roots.
If the discriminant is positive, then there are two distinct roots +, both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number , then the roots are rational—in other cases they may be quadratic irrationals .
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots. The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle.
If the discriminant is positive, then the vertex is not on the -axis but the parabola opens in the direction of the -axis, crossing it twice, so the corresponding equation has two real roots. If the discriminant is negative, then the parabola opens in the opposite direction, never crossing the -axis, and the equation has no ...
The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L , and like the absolute discriminant it indicates which primes are ramified in K / L .
It has the smallest discriminant of all totally real cubic fields, namely 49. [4] The field obtained by adjoining to Q a root of x 3 + x 2 − 3x − 1 is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field. [5]
If the discriminant is zero the fraction converges to the single root of multiplicity two. If the discriminant is positive the equation has two real roots, and the continued fraction converges to the larger (in absolute value) of these. The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ...
In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex ...