Solving Cubic Equations: ax3 +bx2 + cx+d=0 - Essay Sample

Introduction

Cubic equations are defined as the equations having a degree of 3. This suggests that higher exponents are often 3 (Zengin, 2018). In the algebraic formula, the cubic equation's overall form is written in this manner ax3 +bx2 + cx + d = 0, whereby a, b, c, and d are all the numbers with the single limitation which can never be 0 (Zengin, 2018). Therefore, cubic equations may contain only sing term only if they contain three as the exponent. The overall approach used to solve a cubic equation is to decrease it to a quadratic equation (Zengin, 2018). Afterward, you have to resolve the quadratic equation via using the standard means by either using the formula or factorizing.

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When it comes to mathematics, cubic equations have been there all around for years. The early Babylonians had some formulas used to calculate cubic equations (Zengin, 2018). However, archeologists have discovered ancient tablets that were used by the Babylonian, and this shows the tables which aided individuals during that period to calculate the equation and come up with the solution (Biran & Membrez, 2016). This drives back to the 20th BC. This is old. As from that time, some of the mathematicians over and over have further their capability of resolving cubic equations. Currently, most of the methods used to solve cubic equations have been proven. Thanks to every hard-work of the mathematicians over decades (Biran & Membrez, 2016).

For decades now, the cubic equation has proven a vital tool in the advancement of algebraic and geometric mathematics. However, the cubic equation was not invented autonomously; and has since foregone a myriad of analytical tests and simultaneous invention applications before actuating an arbitrary equation (Biran & Membrez, 2016). The purpose of this assignment is to delaminate the proponents of the cubic equation in terms of historical development, exemplification, and subsequent development of the equation, as mentioned above.

Ideally, the cubic equation is a formula incorporated in algebra and trigonometry to solve basic mechanical problems in real life (Biran & Membrez, 2016). The advent of the cubic equation was in early Greece and Egypt. In 430 B.C, Hippocrates, a Greek mathematician, became the first to duplicate a cube through the application of a formula leverages on the cube's line and twice its value in length (Biran & Membrez, 2016). Later, an Egyptian named Menachmus improved on the initial formula and incorporated x2-\-2x-\-3 = x3-Jr3x, a formula to find a right-angled triangle offered that the sum total of its hypotenuse and subsequent area conform into an equilateral cube or square. The Arabs further improved the equation into 1x2+2x2+x+ =0 for conic square solicitation (Biran & Membrez, 2016). Eventually, Tartaglia, a European compounded all previous versions of the equation into one linear equation, possessing divergent coefficients. Hence, the resulting formula was:

ax2+ bx2+cx+d=0

Video Example

https://study.com/academy/lesson/solving-cubic-equations-with-integers.html

The video, in this case, is titled "Solving Cubic equations with integers." Essentially, this video incorporates integers to find solutions to algorithms whose highest power is three. Ideally, I chose this video owing to its articulate short length. On the other hand, the tutor offers a step-by-step analysis of the data formulation procedure. The incorporation of clear fonts further adds to the simplicity of understanding the cubic equation idealized in the video. Furthermore, integers are idealized as the most basic form of all algorithmic computations; hence, this video leveraged on the novice-aspect to highlight imminent problem-solving techniques.

Application

One of the most conspicuous real-world problems that could benefit from the knowledge of the cubic equation is architecture. Normally, curved buildings present design challenges in optimization and eventual applicability (Kim, 2018). Hence, the cubic function would be used to identify sequencing and efficient angle proponents for various house designs. Primarily, this incorporation would be based on the sonic development and splitting of right angles to achieve concise calibrations of the imminent structure (Biran, 2016). Hence, through the rightful application of angle coefficients, the ability to obtain candid positions on curves is possible.

Cubic Equation Example

Solve the following equation:

x2 – 8x2 + 2x + 14 = 0

Part 5-Solving the Equation

Let p(x) = x2 – 8x2 + 2x + 14

Hence, we already know that the possible root values are;

+1, +2, +3, +4, +5, +6, +14

If p (-1) =-1-8-2+14=0

Thus, (x + 1) is a factor of p(x)

– 8x2 + 2x + 14= (x + 1)(x2 – 4x + 14)= (x + 1)(x – 2)(x – 6)

Therefore, the identified roots of the equation are:1, 4, 6

Identifying Mathematical Misconceptions and Obtaining the Meaning

Often, various mathematical problems are idealized as difficult and unnavigable. However, a deeper understanding of the affiliated techniques and formulae does not only bring about more knowledge and understanding but also expands on the realms of more knowledge finding. In essence, the evolution of mathematical formulae compounds to mathematician numerous years of practice and testing (Zengin, 2018). Eventually, the solicitation of knowledge improves unity between mathematics enthusiasts, and overtly promotes advancement in dynamic areas of design. Architecture, for instance, largely borrows from the knowledge of inherent mathematical formulae. Conclusively, the advent of the cubic equation was in early Greece and Egypt. In 430 B.C Hippocrates, a Greek became the first to duplicate a cube through the application of formula leverages on the cube's line and twice its value in length. Later, an Egyptian named Menachmus improved on the initial formula and incorporated the modern arbitrary formula:

ax2+ bx2+cx+ d=0

The subsequent evolution of mathematical formulae further proves that formulae take many forms. The video in this case task, for instance, is titled "Solving Cubic equations with integers." Notably, this video incorporates integers to find solutions to algorithms whose highest power is three. Ideally, the reason for choosing this is attributable to its articulate short length. Nevertheless, the tutor offers a step-by-step analysis of the data formulation procedure. The incorporation of clear fonts further adds to the simplicity of understanding the cubic equation idealized in the video. Furthermore, integers are idealized as the most basic form of all algorithmic computations. In retrospect, the application of the cubic formula spreads far as to assisting architects and designers alike with diverse designs. For instance, curved buildings present design challenges in optimization and eventual applicability. Hence, the cubic function would be used to identify sequencing and efficient angle proponents for various house designs. Primarily, this incorporation would be based on the sonic development and splitting of right angles to achieve concise calibrations of the building plan.

References

Biran, P., &Membrez, C. (2016).The Lagrangian cubic equation. International Mathematics Research Notices, 2016(9), 2569-2631. https://arxiv.org/pdf/1406.6004.pdfhttps://study.com/academy/lesson/solving-cubic-equations-with-integers.html

Kim, H. S., Kim, Y., & Park, S. E. (2018).The reinterpretation and visualization for geometric methods of solving the cubic equation. East Asian mathematical journal, 34(4), 403-427. http://www.koreascience.or.kr/article/JAKO201828458535234.page

Zengin, Y. (2018). Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course. International Journal of Mathematical Education in Science and Technology, 49(7), 1083-1098. https://www.tandfonline.com/doi/abs/10.1080/0020739X.2018.1431850