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The discriminant of the quadratic polynomial + + is , the quantity which appears under the square root in the quadratic formula. If , this discriminant is zero if and only if the polynomial has a double root.
Figure 3. Discriminant signs. In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta: [13] =.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
The roots , of the quadratic polynomial () = + + satisfy + =, =. The first of these equations can be used to find the minimum (or maximum) of P ; see Quadratic equation § Vieta's formulas .
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve ...
The ring [] consists of all roots of all equations x 2 + Bx + C = 0 whose discriminant B 2 − 4C is the product of D by the square of an integer. In particular √ D belongs to Z [ ω ] {\displaystyle \mathbb {Z} [\omega ]} , being a root of the equation x 2 − D = 0 , which has 4 D as its discriminant.
In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of -th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.
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 ...