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The four roots x 1, x 2, x 3, and x 4 for the general quartic equation a x 4 + b x 3 + c x 2 + d x + e = 0 {\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,} with a ≠ 0 are given in the following formula, which is deduced from the one in the section on Ferrari's method by back changing the variables (see § Converting to a depressed quartic ) and ...
The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
The cruciform curve, or cross curve is a quartic plane curve given by the equation = where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a 2 x 2 + b 2 y 2 = 1, and is therefore a rational plane algebraic curve of genus zero.
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points. + + + + = where a ≠ 0.
The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron. With this definition, the circumcenter C of a tetrahedron with vertices x 0, x 1, x 2, x 3 can be formulated as matrix-vector product: [35]
The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is 4fy = x 2, where f is the focal length. At the positive x end of the chord, x = c / 2 and y = d. Since this point is on the parabola, these coordinates must satisfy the equation above.
() = + is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively. The coefficient a is the same value in all three forms. To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2.
He defined the oval as the solution to a differential equation, constructed its subnormals, and again investigated its optical properties. [ 8 ] The French mathematician Michel Chasles discovered in the 19th century that, if a Cartesian oval is defined by two points P and Q , then there is in general a third point R on the same line such that ...