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In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle.The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis.
When the monic quadratic equation with real coefficients is of the form x 2 = c, the general solution described above is useless because division by zero is not well defined. As long as c is positive, though, it is always possible to transform the equation by subtracting a perfect square from both sides and proceeding along the lines ...
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 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]
In mathematics, a quadratic function of a single variable is a function of the form [1] = + +,,where is its variable, and , , and are coefficients.The expression + + , especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two.
Pierre Fermat stated that if p is an odd prime then the equation = + has a solution iff (), and he made similar statement about the equations = +, = +, = and =. x 2 + y 2 , x 2 + 2 y 2 , x 2 − 3 y 2 {\displaystyle x^{2}+y^{2},x^{2}+2y^{2},x^{2}-3y^{2}} and so on are quadratic forms, and the theory of quadratic forms gives a unified way of ...
Solutions of indeterminate quadratic equations (of the type ax 2 + b = y 2). Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century. A cyclic Chakravala method for solving indeterminate equations of ...
The pair (V, Q) consisting of a finite-dimensional vector space V over K and a quadratic map Q from V to K is called a quadratic space, and B as defined here is the associated symmetric bilinear form of Q. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form.