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Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field). A quadratic form in four variables is the equation of a projective surface.
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.
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 quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated ...
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]
The simplest examples of resolvents are X 2 − Δ {\displaystyle X^{2}-\Delta } where Δ {\displaystyle \Delta } is the discriminant , which is a resolvent for the alternating group . In the case of a cubic equation , this resolvent is sometimes called the quadratic resolvent ; its roots appear explicitly in the formulas for the roots of a ...
Private chef Mila Furman shared her top Trader Joe's picks for holiday entertaining. Furman, a Chicago-based chef, has shopped at Trader Joe's for over 20 years. Her recommendations include ...
The graph of a real single-variable quadratic function is a parabola. 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 ...
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