# Solution of cubic and quartic equations pdf

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In algebra , a quartic function is a function of the form.

## Graphical Solution of Cubic and Quartic Equations

In algebra , a quartic function is a function of the form. A quartic equation , or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form. Sometimes the term biquadratic is used instead of quartic , but, usually, biquadratic function refers to a quadratic function of a square or, equivalently, to the function defined by a quartic polynomial without terms of odd degree , having the form.

Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.

The degree four quartic case is the highest degree such that every polynomial equation can be solved by radicals. Lodovico Ferrari is credited with the discovery of the solution to the quartic in , but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately. The Soviet historian I. Depman ru claimed that even earlier, in , Spanish mathematician Valmes was burned at the stake for claiming to have solved the quartic equation.

Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed. The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel—Ruffini theorem in , proving that all attempts at solving the higher order polynomials would be futile.

Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus. It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics , computer-aided design , computer-aided manufacturing and optics. Here are examples of other geometric problems whose solution involves solving a quartic equation. In computer-aided manufacturing , the torus is a shape that is commonly associated with the endmill cutter.

To calculate its location relative to a triangulated surface, the position of a horizontal torus on the z -axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated. A quartic equation arises also in the process of solving the crossed ladders problem , in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.

In optics, Alhazen's problem is " Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer. Finding the distance of closest approach of two ellipses involves solving a quartic equation. The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation.

An example arises in the Timoshenko-Rayleigh theory of beam bending. Intersections between spheres, cylinders, or other quadrics can be found using quartic equations. Letting F and G be the distinct inflection points of the graph of a quartic function, and letting H be the intersection of the inflection secant line FG and the quartic, nearer to G than to F , then G divides FH into the golden section : [15].

Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.

The possible cases for the nature of the roots are as follows: [16]. There are some cases that do not seem to be covered, but they cannot occur. The four roots x 1 , x 2 , x 3 , and x 4 for the general quartic equation.

Such a factorization will take one of two forms:. In either case, the roots of Q x are the roots of the factors, which may be computed using the formulas for the roots of a quadratic function or cubic function. Detecting the existence of such factorizations can be done using the resolvent cubic of Q x.

It turns out that:. In fact, several methods of solving quartic equations Ferrari's method , Descartes' method , and, to a lesser extent, Euler's method are based upon finding such factorizations. Then the roots of our quartic Q x are. For solving purposes, it is generally better to convert the quartic into a depressed quartic by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable.

This depressed quartic can be solved by means of a method discovered by Lodovico Ferrari. The depressed equation may be rewritten this is easily verified by expanding the square and regrouping all terms in the left-hand side as. After regrouping the coefficients of the power of y on the right-hand side, this gives the equation. As the value of m may be arbitrarily chosen, we will choose it in order to complete the square on the right-hand side.

This implies that the discriminant in y of this quadratic equation is zero, that is m is a root of the equation. This is the resolvent cubic of the quartic equation. The value of m may thus be obtained from Cardano's formula. When m is a root of this equation, the right-hand side of equation 1 is the square. This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients.

Therefore, equation 1 may be rewritten as. This equation is easily solved by applying to each factor the quadratic formula.

Solving them we may write the four roots as. Descartes [19] introduced in the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. By equating coefficients , this results in the following system of equations:. One can now eliminate both t and v by doing the following:. If u is a square root of a non-zero root of this resolvent such a non-zero root exists except for the quartic x 4 , which is trivially factored ,.

The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of u for the square root of U merely exchanges the two quadratics with one another.

A variant of the previous method is due to Euler. Observe that, if. This is indeed true and it follows from Vieta's formulas. For the same reason,. Therefore, the numbers r 1 , r 2 , r 3 , and r 4 are such that. But a straightforward computation shows that. The symmetric group S 4 on four elements has the Klein four-group as a normal subgroup. This suggests using a resolvent cubic whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots; see Lagrange resolvents for the general method.

If we set. These are the roots of the polynomial. Substituting the s i by their values in term of the x i , this polynomial may be expanded in a polynomial in s whose coefficients are symmetric polynomials in the x i. By the fundamental theorem of symmetric polynomials , these coefficients may be expressed as polynomials in the coefficients of the monic quartic. This polynomial is of degree six, but only of degree three in s 2 , and so the corresponding equation is solvable by the method described in the article about cubic function.

By substituting the roots in the expression of the x i in terms of the s i , we obtain expression for the roots.

In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the x i.

These expressions are unnecessarily complicated, involving the cubic roots of unity , which can be avoided as follows. If s is any non-zero root of 3 , and if we set. We therefore can solve the quartic by solving for s and then solving for the roots of the two factors using the quadratic formula. This gives exactly the same formula for the roots as the one provided by Descartes' method.

There is an alternative solution using algebraic geometry [23] In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three reducible quadratic curves pairs of lines that pass through these points this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents , and then use these linear equations to solve the quadratic.

Retrieved 27 July Beckmann Zoll American Mathematical Monthly. Theory of Vibration: An Introduction. The American Mathematical Monthly. Journal of Symbolic Computation. Mathematics Magazine. Polynomials and polynomial functions. Categories : Elementary algebra Equations Polynomial functions.

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## Quartic function

Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. SOME years ago you published some interesting communications in regard to the graphical solution of cubic and quartic equations vol. The solutions then given give only the real roots of the equation.

Хейл выжидал. Стояла полная тишина, и он внимательно прислушался. Ничего. Вроде бы на нижней ступеньке никого. Может, ему просто показалось. Какая разница, Стратмор никогда не решится выстрелить, пока он прикрыт Сьюзан. Но когда он начал подниматься на следующую ступеньку, не выпуская Сьюзан из рук, произошло нечто неожиданное.

Likely you are familiar with how to solve a quadratic equation. Given a quadratic On the other hand, the cubic formula is quite a bit messier. The polynomial x4.

## What you should know about cubic and quartic equations

Мысли Стратмора судорожно метались в поисках решения. Всегда есть какой-то выход. Наконец он заговорил - спокойно, тихо и даже печально: - Нет, Грег, извини.

Я, пожалуй, занесу его в полицейский участок по пути в… - Perdon, - прервал его Ролдан, занервничав.  - Я мог бы предложить вам более привлекательную идею.  - Ролдан был человек осторожный, а визит в полицию мог превратить его клиентов в бывших клиентов.

Похоже, что-то стряслось, - сказала Сьюзан.  - Наверное, увидел включенный монитор. - Черт возьми! - выругался коммандер.  - Вчера вечером я специально позвонил дежурному лаборатории систем безопасности и попросил его сегодня не выходить на работу.

### Solution of Cubic and Quartic Equations

Код ошибки 22. Она попыталась вспомнить, что это. Сбои техники в Третьем узле были такой редкостью, что номера ошибок в ее памяти не задерживалось. Сьюзан пролистала справочник и нашла нужный список. 19: ОШИБКА В СИСТЕМНОМ РАЗДЕЛЕ 20: СКАЧОК НАПРЯЖЕНИЯ 21: СБОЙ СИСТЕМЫ ХРАНЕНИЯ ДАННЫХ Наконец она дошла до пункта 22 и, замерев, долго всматривалась в написанное. Потом, озадаченная, снова взглянула на монитор.

Он… это кольцо… он совал его нам в лицо, тыкал своими изуродованными пальцами. Он все протягивал к нам руку - чтобы мы взяли кольцо. Я не хотела брать, но мой спутник в конце концов его. А потом этот парень умер.

- Может, вы знаете имя этой женщины. Клушар некоторое время молчал, потом потер правый висок. Он был очень бледен. - Н-нет… Не думаю… - Голос его дрожал. Беккер склонился над. - Вам плохо. Клушар едва заметно кивнул: - Просто… я переволновался, наверное.

Это ловушка. Энсей Танкадо всучил вам Северную Дакоту, так как он знал, что вы начнете искать. Что бы ни содержалось в его посланиях, он хотел, чтобы вы их нашли, - это ложный след. - У тебя хорошее чутье, - парировал Стратмор, - но есть кое-что. Я ничего не нашел на Северную Дакоту, поэтому изменил направление поиска.

Мидж как ни чем не бывало стояла в приемной возле двойной двери директорского кабинета и протягивала к нему руку ладонью вверх. - Ключ, Чед. Бринкерхофф покраснел до корней волос и повернулся к мониторам. Ему хотелось чем-то прикрыть эти картинки под потолком, но. Он был повсюду, постанывающий от удовольствия и жадно слизывающий мед с маленьких грудей Кармен Хуэрты.

#### COMMENT 2

• The solution of cubic and quartic equations. In the 16th century in Italy, there occurred the first progress on polynomial equations beyond the quadratic case. Eilal A. - 08.05.2021 at 09:47
• Solution of Cubic and Quartic Equations presents the classical methods in solving cubic and quartic equations to the highest possible degree of efficiency. Shauncey A. - 09.05.2021 at 11:54