Exact solutions of sixth and fifth degree equations

The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

Thermodynamic Study of Adsorption of Tamoxifen Drug on Activated Nano-Coal Particles prepared from Stem of Eucalyptus and its Diagnosis by different Techniques

The research included preparing nano-activated charcoal from the stem of the eucalyptus trees. The prepared nano-coal particles were diagnosed using various techniques, such as, transmission electron microscopy (TEM), and energy-dispersive X-ray (EDX) technology. The research included a spectroscopic study of the adsorption of tamoxifen on the prepared nano-coal, and the results showed that the appropriate concentration is 2.5 × 10-5 molar, with an appropriate weight of the adsorbent material which is 0.4-gram, the time of equilibrium is 40 minutes, and the percentage of adsorption efficiency increases with passing time, the adsorption process is dependent to a second-degree equation, according to the correlation coefficient (R2), which gave the value of 0.9999. Thermodynamic functions were calculated for the adsorption process at different temperatures (17.5, 27.5, 37.5, and 47.5ºC), and it was found that the adsorption process is a heat emission process (ΔH negative), the adsorption is physical because it is less than 40 kJ/mol, and the free energy is negative (ΔG), and entropy adsorption was negative (ΔS), meaning less random.

Análise praxeológica das práticas de resolução de equações do primeiro grauPragmatic analysis of first-degree equation resolution practices

Neste trabalho apresentamos de forma sucinta os resultados de uma pesquisa de mestrado que teve como objeto o estudo da resolução de equações redutíveis à equação do primeiro grau, sob a perspectiva teórico-metodológica da TAD – Teoria Antropológica do Didático. Nessa investigação engendrou-se um Percurso de Estudo e Pesquisa (PEP), que nos levou ao encontro de quatro diferentes práticas de resolução de equações do primeiro grau em diferentes épocas. As análises dessas práticas foram realizadas pelo contraste com o Modelo Epistemológico de Referência (MER) adotado, construído para colocar em evidência as propriedades operatórias enquanto regras para resolver equações do primeiro grau. O MER foi construído utilizando as noções de anéis e corpos, mais especificamente, do corpo dos inteiros módulo . Resultados neste percurso nos apontam que as técnicas históricas estão mais próximas das propriedades operatórias e são preferíveis frente a algumas técnicas atuais.

Negociações, rupturas e renegociações numa sala de aula com o saber equação do segundo grauNegotiations, breaks and renegotiations in a classroom with the knowledge of the second-degree equation

O objetivo do artigo é analisar as relações contratuais, que emergem da relação didática, numa sala de aula de matemática no 9º ano do ensino fundamental, quando o saber algébrico é a equação do segundo grau entra em cena no jogo didático. Para tornar possível a investigação, elegemos como campo teórico e metodológico a noção de Contrato Didático, um dos pilares da teoria das situações didáticas, esse fenômeno implica regras que emergem da relação didática e que são determinantes para o funcionamento do sistema didático. Na metodologia utilizamos uma filmagem e, posteriormente, a transcrição e análise. Os resultados apontam que o professor contempla, negociações, rupturas e renegociações na relação didática. No entanto, podemos observar em alguns momentos uma reorganização contratual e o estabelecimento de um núcleo duro do contrato.

The accurate calculation of the geodetic heights of the celestial body’s surface points relative to the triaxial ellipsoid

A sphere or ellipsoid of revolution are usually used for approximation of the physical surface of the Earth. In some cases, a triaxial ellipsoid is used. The calculation of the geodetic height of points on the Earth’s surface is carried out mainly by approximate methods using the formulas for the dependence of spatial rectangular coordinates x, y, z on geodesics B, L, H. However, there are small bodies of the Solar system, for example, Eros 433 asteroid, for which such variants of the first approximation are incorrect, since in this case both first approximations are not small quantities. This article proposes a fundamentally new approach to calculating the geodesic height relative to the triaxial ellipsoid, based on the joint use of the normal equation for a surface passing through a given point and the equations of the surface itself. The method is reduced to solving the sixth-degree equation by the Sturm method and the fourth-degree equation by the Ferrari method.

Analysis of Strata Behavior Process Characteristics of Gob-Side Entry Retaining with Roof Cutting and Pressure Releasing Based on Composite Roof Structure

In order to explore the characteristics of rock pressure distribution with roof cutting and pressure releasing under different composite roof structures and optimize the support design of entry retaining, the mechanical analysis and numerical simulation are used to analyze the structure characteristics of composite roof and the effect of roof cutting under composite roof in this paper. Besides, taking the 8304 working face of Tashan Coal Mine as an example, the results of theoretical research are verified by field-monitoring data of hydraulic supports, working resistance, and roadway deformation. The results show that the weak interlayer in the composite roof is easily damaged under the external force and the distribution of the layer has a key effect on the roof characteristics. When the weak interlayer is located at the middle of the roof cutting layer range, the demand of the roadway support strength is the highest; when the weak interlayer is located at the top of the roof cutting layer range, the demand of the roadway support strength is the lowest. Furthermore, with the increase of the height of the weak interlayer in the roof cutting layer range, the stress concentration peak of the coal wall side decreases first and then rises, then descends again, and the trend can be fitted by the curve of a three-degree equation.

Static Range of Motion of the First Metatarsal in the Sagittal and Frontal Planes

The first metatarsal and medial cuneiform form an important functional unit in the foot, called “first ray”. The first ray normal range of motion (ROM) is difficult to quantify due to the number of joints that are involved. Several methods have previously been proposed. Controversy exists related to normal movement of the first ray frontal plane accompanying that in the sagittal plane. The objective of this study was to investigate the ROM of the first ray in the sagittal and frontal planes in normal feet. Anterior-posterior radiographs were done of the feet of 40 healthy participants with the first ray in a neutral position, maximally dorsiflexed and maximally plantarflexed. They were digitalized and the distance between the tibial malleolus and the intersesamoid crest in the three positions mentioned was measured. The rotation of the first ray in these three positions was measured. A polynomic function that fits a curve describing the movement observed in the first ray was obtained using the least squares method. ROM of the first ray in the sagittal plane was 6.47 (SD 2.59) mm of dorsiflexion and 6.12 (SD 2.55) mm of plantarflexion. ROM in the frontal plane was 2.69 (SD 4.03) degrees of inversion during the dorsiflexion and 2.97 (SD 2.72) degrees during the plantarflexion. A second-degree equation was obtained, which represents the movement of the first ray. Passive dorsiflexion and plantarflexion of the first ray were accompanied by movements in the frontal plane: 0.45 degrees of movement were produced in the frontal plane for each millimeter of displacement in the sagittal plane. These findings might be useful for the future design of instruments for clinically quantifying first ray mobility.