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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.
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
A natural extension is to consider Boolean formulas of word equations, [4] in which also negation and disjunction is allowed. In fact, every system (and even every Boolean formula) of word equations, is equivalent to a single word equation. [4] Thus, many results on word equations generalise immediately to such systems (resp. 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 ...
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
Quadratic forms are not to be confused with quadratic equations, which have only one variable and may include terms of degree less than two. A quadratic form is a specific instance of the more general concept of forms .
The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is x(x − s) + (y − 1)(y − p) = 0. The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle)
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).