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  2. Discriminant - Wikipedia

    en.wikipedia.org/wiki/Discriminant

    If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer k ≤ n/4 such that there are 2k pairs of complex conjugate roots and n − 4k real roots. If the discriminant is negative, the number of non-real roots is not a multiple of 4.

  3. Quadratic equation - Wikipedia

    en.wikipedia.org/wiki/Quadratic_equation

    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. If the discriminant is zero, then there is exactly one real root − b 2 a , {\displaystyle -{\frac {b}{2a}},} sometimes called a repeated or double root or two equal roots.

  4. Quadratic formula - Wikipedia

    en.wikipedia.org/wiki/Quadratic_formula

    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.

  5. Quadratic integer - Wikipedia

    en.wikipedia.org/wiki/Quadratic_integer

    In particular √ D belongs to [], being a root of the equation x 2 − D = 0, which has 4D as its discriminant. The square root of any integer is a quadratic integer, as every integer can be written n = m 2 D, where D is a square-free integer, and its square root is a root of x 2 − m 2 D = 0.

  6. Discriminant of an algebraic number field - Wikipedia

    en.wikipedia.org/wiki/Discriminant_of_an...

    An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant. [3] Cyclotomic fields: let n > 2 be an integer, let ζ n be a primitive nth root of unity, and let K n = Q(ζ n) be the nth cyclotomic field. The discriminant of K n is given by [2] [4]

  7. Vieta's formulas - Wikipedia

    en.wikipedia.org/wiki/Vieta's_formulas

    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 .

  8. Casus irreducibilis - Wikipedia

    en.wikipedia.org/wiki/Casus_irreducibilis

    Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if ⁠ q 2 / 4 ⁠ + ⁠ p 3 / 27 ⁠ < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the ...

  9. Quadratic function - Wikipedia

    en.wikipedia.org/wiki/Quadratic_function

    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 ...