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He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations. [18] In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x 3 + 2x 2 + 10x = 20.
A cubic function with real coefficients has either one or three real roots (which may not be distinct); [1] all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum.
The Darboux cubic is the locus of a point X such that X* is on the line LX, where L is the de Longchamps point. Also, this cubic is the locus of X such that the pedal triangle of X is the cevian triangle of some point (which lies on the Lucas cubic). Also, this cubic is the locus of a point X such that the pedal triangle of X and the anticevian ...
The first degree polynomial equation = + is a line with slope a. A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results:
Tucker cubic (cubic K011 in the Catalogue) of triangle ABC drawn using the GeoGebra command Cubic(A,B,C,11). GeoGebra, the software package for interactive geometry, algebra, statistics and calculus application has a built-in tool for drawing the cubics listed in the Catalogue. [3] The command Cubic( <Point>, <Point>, <Point>, n)
Tschirnhausen cubic, case of a = 1. In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation = where sec is the secant function.
The decreasing part of the curve to the right of point C in Fig. 1 describes a gas, while the decreasing part to the left of point E describes a liquid. These two parts are separated by a region between the local minimum and local maximum on the curve with positive slope that violates the stability criterion.
Descartes theory of geometric solution of equations uses a parabola to introduce cubic equations, in this way it is possible to set up an equation whose solution is a cube root of two. Note that the parabola itself is not constructible except by three dimensional methods.