Related papers: A Simple Proof of the Quadratic Formula
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
In this paper, we derive the quadratic formula as a consequence of constructively proving the existence of standard and factored forms for general form real quadratic functions. Emphasis is put on connections to graphing of corresponding…
The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.
After Abel Ruffini theorem and Galois Theory the search for a method or formula to solve quintic equation ends. This paper discuss about the radical solution of quintic equation using a method that could be proved in some simple steps. A…
We give a short constructive proof for the existence and uniqueness of the rational normal form of a quadratic matrix.
This article discusses some important applications of the quadratic function with the aim of highlighting the importance of cuadr\'aticas.- forms are also intended to show how a simple function covers virtually all areas of knowledge are…
This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the…
In this paper we show that Cardanos formula for the solution of cubic equations can be reduced to expressions involving only square roots if the real root is rational.
We give a simple direct proof of Fermat's two squares theorem. Our argument uses no intricate notions or ideas; one might say that it is a proof by careful bookkeeping. As such, the proof may be particularly easy to comprehend by students…
Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of…
In this note we shall give a new proof to a quadrature formulae due to Newton.
It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new…
We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.
The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.
The shortest known proof of the law of quadratic reciprocity (without supplements) is presented.
We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
Iterating Newton's method symbolically for the general quadratic yields a rational function, the numerator and denominator of which are polynomials with highly composite coefficients.