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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.
The roots, stationary points, inflection point and concavity of a cubic polynomial x 3 − 6x 2 + 9x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives. The critical points of a cubic function are its stationary points, that is the points where the slope of the function is zero. [2]
The real points of a non-singular projective cubic fall into one or two 'ovals'. One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not ...
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:
The nine intersections of = and = (). In mathematics, Cramer's paradox or the Cramer–Euler paradox [1] is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve.
The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
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.
The parameter t can be eliminated easily giving the Cartesian equation 27 a y 2 = ( a − x ) ( 8 a + x ) 2 {\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}} . If the curve is translated horizontally by 8 a and the signs of the variables are changed, the equations of the resulting right-opening curve are