Quartic equation - Citizendia

In mathematics, a quartic equation is one which can be expressed as a quartic function equalling zero. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A quartic function is a function of the form f(x=ax^4+bx^3+cx^2+dx+e \ with nonzero a; or in other words a Polynomial The general form of a quartic equation is

$ax^4+bx^3+cx^2+dx+e=0 \,$

where $a\ne 0$ .

The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i. In Mathematics, an n th root of a Number a is a number b such that bn = a. e. , one where the coefficients can take any value).

History

Quartic equations were first considered by Jaina Mathematicians in ancient India between 400 BC and 200 AD. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. This article deals with the geophysical region in Asia For geopolitical treatments see South Asia. Events By place Persian Empire Artaxerxes II King of Persia appoints Tissaphernes to take over all the districts in Events By Place World Human population reaches about 257 million See History of Cubic equation for more details. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve.

Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it couldn't be published immediately. Lodovico Ferrari ( February 2, 1522 – October 5, 1565) was an Italian Mathematician. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. [1] The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna (1545). The Ars Magna ( Latin: "The Great Art" is an important book on Algebra written by Gerolamo Cardano.

The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to Polynomial equations of The notes left by Évariste Galois in 1832 later led to the complete theory of the roots of polynomials, of which this theorem was one result. ^[1]

Applications

Polynomials of high degrees often appear in problems involving optimization, and sometimes these polynomials happen to be quartics, but this is a coincidence. In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function

Quartics often arise in computer graphics and during ray-tracing against surfaces such as quadrics or tori surfaces, which are the next level beyond the sphere and developable surfaces. In Computer graphics, ray tracing is a technique for generating an image by tracing the path of Light through pixels in an Image plane In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, a developable surface is a Surface with zero Gaussian curvature. [2]

Another frequent generator of quartics is the intersection of two ellipses.

In Computer-aided manufacturing, the torus is a common shape associated with the endmill cutter. Computer-aided manufacturing ( CAM) is the use of computer-based software tools that assist engineers and machinists in manufacturing or prototyping product components In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar An endmill is a type of Milling cutter, a cutting tool used in industrial milling applications In order 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. Over 10% of the computational time in a CAM system can be consumed simply calculating the solution to millions of quartic equations.

A program demonstrating various analytic solutions to the quartic was provided in Graphics Gems Book V. [3] However, none of the three algorithms implemented are unconditionally stable. In an updated version of the paper[4], which compares the 3 algorithms from the original paper and 2 others, it is demonstrated that computationally stable solutions exist only for 4 of the possible 16 sign combinations of the quartic coefficients.

Solving a quartic equation

Special cases

Consider a quartic equation expressed in the form $a 0 x 4 + a 1 x 3 + a 2 x 2 + a 3 x + a 4 = 0$ :

Degenerate case

If a₄ (the constant term) = 0, then one of the roots is x = 0, and the other roots can be found by dividing by x, and solving the resulting cubic equation,

$a_0x^3+a_1x^2+a_2x+a_3=0. \,$

Evident roots: 1 and -1 and -k

Call our quartic polynomial Q(x). for the degeneracy of a Graph, see Arboricity#Related_concepts. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Since 1 raised to any power is 1, $Q (1) = a 0 + a 1 + a 2 + a 3 + a 4$ . Thus if $a 0 + a 1 + a 2 + a 3 + a 4 = 0$ , Q(1)=0 and so x=1 is a root of Q(x). It can similarly be shown that if $a 0 + a 2 + a 4 = a 1 + a 3$ , x=-1 is a root.

In either case the full quartic can then be divided by the factor (x-1) or (x+1) respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve.

If $a 1 = a 0 k$ , $a 2 = 0$ and $a 4 = a 3 k$ , then x= - k is a root of the equation. The full quartic can then be factorized this way:

$a_0 x^4+ a_0 k x^3 + a_3 x + a_3 k = a_0 x^3 (x +k ) + a_3 (x+k) = (a_0 x^3 + a_3) (x+k) \,$ .

If $a 1 = a 0 k$ , $a 3 = a 2 k$ and $a 4 = 0$ , x=0 and x= -k are two known roots. Q(x)divided by $x (x + k)$ is a quadratic equation. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree.

Biquadratic equations

A quartic equation where a₃ and a₁ are equal to 0 takes the form

$a_0x^4+a_2x^2+a_4=0\,\!$

and thus is a biquadratic equation, which is easy to solve: let $z = x 2$ , so our equation turns to

$a_0z^2+a_2z+a_4=0\,\!$

which is a simple quadratic equation, whose solutions are easily found using the quadratic formula:

$z={{-a_2\pm\sqrt{a_2^2-4a_0a_4}} \over {2a_0}}\,\!$

When we've solved it (i. e. found these two z values), we can extract x from them

$x_1=+\sqrt{z_+}\,\!$

$x_2=-\sqrt{z_+}\,\!$

$x_3=+\sqrt{z_-}\,\!$

$x_4=-\sqrt{z_-}\,\!$

If either of the z solutions were negative or complex numbers, some of the x solutions are complex numbers.

Symmetric equations

Symmetric equations $a 0 x 4 + a 1 x 3 + a 2 x 2 + a 1 x + a 0 = 0$ are particular cases of the quasi symmetric equations. So are the biquadratic equations or equations like these ones:

$a_0x^4+a_1x^3+a_2x^2-a_1x+a_0=0 \,$

$a_0x^4 + k a_0=0 \,$

Quasi Symmetric equations

$a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2=0 \,$

Steps:

1) Divide by $x 2$

2) Use variable change $z=x + m/x. \,$ .

The general case, along Ferrari's lines

To begin, the quartic must first be converted to a depressed quartic.

Converting to a depressed quartic

Let

$A x^4 + B x^3 + C x^2 + D x + E = 0 \qquad\qquad(1')$

be the general quartic equation which it is desired to solve. Divide both sides by A,

$x^4 + {B \over A} x^3 + {C \over A} x^2 + {D \over A} x + {E \over A} = 0.$

The first step should be to eliminate the x³ term. To do this, change variables from x to u, such that

$x = u - {B \over 4 A}$ .

Then

$\left( u - {B \over 4 A} \right)^4 + {B \over A} \left( u - {B \over 4 A} \right)^3 + {C \over A} \left( u - {B \over 4 A} \right)^2 + {D \over A} \left( u - {B \over 4 A} \right) + {E \over A} = 0.$

Expanding the powers of the binomials produces

$\left( u^4 - {B \over A} u^3 + {6 u^2 B^2 \over 16 A^2} - {4 u B^3 \over 64 A^3} + {B^4 \over 256 A^4} \right) + {B \over A} \left( u^3 - {3 u^2 B \over 4 A} + {3 u B^2 \over 16 A^2} - {B^3 \over 64 A^3} \right) + {C \over A} \left( u^2 - {u B \over 2 A} + {B^2 \over 16 A^2} \right) + {D \over A} \left( u - {B \over 4 A} \right) + {E \over A} = 0.$

Collecting the same powers of u yields

$u^4 + \left( {-3 B^2 \over 8 A^2} + {C \over A} \right) u^2 + \left( {B^3 \over 8 A^3} - {B C \over 2 A^2} + {D \over A} \right) u + \left( {-3 B^4 \over 256 A^4} + {C B^2 \over 16 A^3} - {B D \over 4 A^2} + {E \over A} \right) = 0.$

Now rename the coefficients of u. Let

$\begin{align}\alpha & = {-3 B^2 \over 8 A^2} + {C \over A} ,\\\beta & = {B^3 \over 8 A^3} - {B C \over 2 A^2} + {D \over A} ,\\\gamma & = {-3 B^4 \over 256 A^4} + {C B^2 \over 16 A^3} - {B D \over 4 A^2} + {E \over A} .\end{align}$

The resulting equation is

$u^4 + \alpha u^2 + \beta u + \gamma = 0 \qquad \qquad (1)$

which is a depressed quartic equation.

If $\beta=0 \$ then we have a Biquadratic equation, which (as explained above) is easily solved; using reverse substitution we can find our values for $x$ . In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero

If $\gamma=0 \$ then one of the roots is $u=0 \$ and the other roots can be found by dividing by $u$ , and solving the resulting depressed cubic equation,

$u^3 + \alpha u + \beta = 0 \ ;$

using reverse substitution we can find our values for $x$ . This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve.

Ferrari's solution

Otherwise, the depressed quartic can be solved by means of a method discovered by Lodovico Ferrari. Lodovico Ferrari ( February 2, 1522 – October 5, 1565) was an Italian Mathematician. Once the depressed quartic has been obtained, the next step is to add the valid identity

$(u^2 + \alpha)^2 - u^4 - 2 \alpha u^2 = \alpha^2\,$

to equation (1), yielding

$(u^2 + \alpha)^2 + \beta u + \gamma = \alpha u^2 + \alpha^2. \qquad \qquad (2)$

The effect has been to fold up the u⁴ term into a perfect square: (u² + α)². This article refers to the REM live recording For the mathematical term see Perfect square. The second term, αu² did not disappear, but its sign has changed and it has been moved to the right side.

The next step is to insert a variable y into the perfect square on the left side of equation (2), and a corresponding 2y into the coefficient of u² in the right side. To accomplish these insertions, the following valid formulas will be added to equation (2),

$\begin{matrix} (u^2+\alpha+y)^2-(u^2+\alpha)^2 & = & 2y(u^2+\alpha)+ y^2\ \ \\ & = & 2yu^2+2y\alpha+y^2, \end{matrix}$

and

$0 = (\alpha + 2 y) u^2 - 2 y u^2 - \alpha u^2\,$

These two formulas, added together, produce

$(u^2 + \alpha + y)^2 - (u^2 + \alpha)^2 = (\alpha + 2 y) u^2 - \alpha u^2 + 2 y \alpha + y^2 \qquad \qquad (y-\hbox{insertion})\,$

which added to equation (2) produces

$(u^2 + \alpha + y)^2 + \beta u + \gamma = (\alpha + 2 y) u^2 + (2 y \alpha + y^2 + \alpha^2).\,$

This is equivalent to

$(u^2 + \alpha + y)^2 = (\alpha + 2 y) u^2 - \beta u + (y^2 + 2 y \alpha + \alpha^2 - \gamma). \qquad \qquad (3)\,$

The objective now is to choose a value for y such that the right side of equation (3) becomes a perfect square. This can be done by letting the discriminant of the quadratic function become zero. To explain this, first expand a perfect square so that it equals a quadratic function:

$(s u + t)^2 = (s^2) u^2 + (2 s t) u + (t^2).\,$

The quadratic function on the right side has three coefficients. It can be verified that squaring the second coefficient and then subtracting four times the product of the first and third coefficients yields zero:

$(2 s t)^2 - 4 (s^2) (t^2) = 0.\,$

Therefore to make the right side of equation (3) into a perfect square, the following equation must be solved:

$(-\beta)^2 - 4 (2 y + \alpha) (y^2 + 2 y \alpha + \alpha^2 - \gamma) = 0.\,$

Multiply the binomial with the polynomial,

$\beta^2 - 4 (2 y^3 + 5 \alpha y^2 + (4 \alpha^2 - 2 \gamma) y + (\alpha^3 - \alpha \gamma)) = 0\,$

Divide both sides by −4, and move the −β²/4 to the right,

$2 y^3 + 5 \alpha y^2 + ( 4 \alpha^2 - 2 \gamma ) y + \left( \alpha^3 - \alpha \gamma - {\beta^2 \over 4} \right) = 0 \qquad \qquad$

This is a cubic equation for y. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Divide both sides by 2,

$y^3 + {5 \over 2} \alpha y^2 + (2 \alpha^2 - \gamma) y + \left( {\alpha^3 \over 2} - {\alpha \gamma \over 2} - {\beta^2 \over 8} \right) = 0. \qquad \qquad (4)$

Conversion of the nested cubic into a depressed cubic

Equation (4) is a cubic equation nested within the quartic equation. It must be solved in order to solve the quartic. To solve the cubic, first transform it into a depressed cubic by means of the substitution

$y = v - {5 \over 6} \alpha.$

Equation (4) becomes

$\left( v - {5 \over 6} \alpha \right)^3 + {5 \over 2} \alpha \left( v - {5 \over 6} \alpha \right)^2 + (2 \alpha^2 - \gamma) \left( v - {5 \over 6} \alpha \right) + \left( {\alpha^3 \over 2} - {\alpha \gamma \over 2} - {\beta^2 \over 8} \right) = 0.$

Expand the powers of the binomials,

$\left( v^3 - {5 \over 2} \alpha v^2 + {25 \over 12} \alpha^2 v - {125 \over 216} \alpha^3 \right) + {5 \over 2} \alpha \left( v^2 - {5 \over 3} \alpha v + {25 \over 36} \alpha^2 \right) + (2 \alpha^2 - \gamma) v - {5 \over 6} \alpha (2 \alpha^2 - \gamma ) + \left( {\alpha^3 \over 2} - {\alpha \gamma \over 2} - {\beta^2 \over 8} \right) = 0.$

Distribute, collect like powers of v, and cancel out the pair of v² terms,

$v^3 + \left( - {\alpha^2 \over 12} - \gamma \right) v + \left( - {\alpha^3 \over 108} + {\alpha \gamma \over 3} - {\beta^2 \over 8} \right) = 0.$

This is a depressed cubic equation.

Relabel its coefficients,

$P = - {\alpha^2 \over 12} - \gamma,$

$Q = - {\alpha^3 \over 108} + {\alpha \gamma \over 3} - {\beta^2 \over 8}.$

The depressed cubic now is

$v^3 + P v + Q = 0. \qquad \qquad (5)$

Solving the nested depressed cubic

The solutions (any solution will do, so pick any of the three complex roots) of equation (5) are

let $U=\sqrt[3]{{Q\over 2}\pm \sqrt{{Q^{2}\over 4}+{P^{3}\over 27}}}$

(taken from Cubic equation)

$v = {P\over 3U} - U$

therefore the solution of the original nested cubic is

$y = - {5 \over 6} \alpha + {P\over 3U} - U \qquad \qquad (6)$

Remember 1: $P=0 \Longleftarrow {Q\over 2} + \sqrt{{Q^{2}\over 4}+{P^{3}\over 27}}=0$

Remember 2: $\lim_{P\to 0}{P \over \sqrt[3]{{Q\over 2} + \sqrt{{Q^{2}\over 4}+{P^{3}\over 27}}}}=0$

Folding the second perfect square

With the value for y given by equation (6), it is now known that the right side of equation (3) is a perfect square of the form

$(s^2)u^2+(2st)u+(t^2) = \left(\left(\sqrt{(s^2)}\right)u + {(2st) \over 2\sqrt{(s^2)}}\right)^2$

(This is correct for both signs of square root, as long as the same sign is taken for both square roots. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. A ± is redundant, as it would be absorbed by another ± a few equations further down this page. )

so that it can be folded:

$(\alpha + 2 y) u^2 + (- \beta) u + (y^2 + 2 y \alpha + \alpha^2 - \gamma ) = \left( \left(\sqrt{(\alpha + 2y)}\right)u + {(-\beta) \over 2\sqrt{(\alpha + 2 y)}} \right)^2$ .

Note: If β ≠ 0 then α + 2y ≠ 0. If β = 0 then this would be a biquadratic equation, which we solved earlier.

Therefore equation (3) becomes

$(u^2 + \alpha + y)^2 = \left( \left(\sqrt{\alpha + 2 y}\right)u - {\beta \over 2\sqrt{\alpha + 2 y}} \right)^2 \qquad\qquad (7)$ .

Equation (7) has a pair of folded perfect squares, one on each side of the equation. The two perfect squares balance each other.

If two squares are equal, then the sides of the two squares are also equal, as shown by:

$(u^2 + \alpha + y) = \pm\left( \left(\sqrt{\alpha + 2 y}\right)u - {\beta \over 2\sqrt{\alpha + 2 y}} \right) \qquad\qquad (7')$ .

Collecting like powers of u produces

$u^2 + \left(\mp_s \sqrt{\alpha + 2 y}\right)u + \left( \alpha + y \pm_s {\beta \over 2\sqrt{\alpha + 2 y}} \right) = 0 \qquad\qquad (8)$ .

Note: The subscript s of $\pm_s$ and $\mp_s$ is to note that they are dependent.

Equation (8) is a quadratic equation for u. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Its solution is

$u={\pm_s\sqrt{\alpha + 2 y} \pm_t \sqrt{(\alpha + 2y) - 4(\alpha + y \pm_s {\beta \over 2\sqrt{\alpha + 2 y}})} \over 2}.$

Simplifying, one gets

$u={\pm_s\sqrt{\alpha + 2 y} \pm_t \sqrt{-\left(3\alpha + 2y \pm_s {2\beta \over \sqrt{\alpha + 2 y}} \right)} \over 2}.$

This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are

$x=-{B \over 4A} + {\pm_s\sqrt{\alpha + 2 y} \pm_t \sqrt{-\left(3\alpha + 2y \pm_s {2\beta \over \sqrt{\alpha + 2 y}} \right)} \over 2}. \qquad\qquad (8')$

Remember: The two $\pm_s$ come from the same place in equation (7'), and should both have the same sign, while the sign of $\pm_t$ is independent.

Summary of Ferrari's method

Given the quartic equation

A x 4 + B x 3 + C x 2 + D x + E = 0,

its solution can be found by means of the following calculations:

$\alpha = - {3 B^2 \over 8 A^2} + {C \over A},$

$\beta = {B^3 \over 8 A^3} - {B C \over 2 A^2} + {D \over A},$

$\gamma = {-3 B^4 \over 256 A^4} + {C B^2 \over 16 A^3} - {B D \over 4 A^2} + {E \over A}.$

If $\,\beta=0,$ then

$x=-{B\over 4A}\pm_s\sqrt{-\alpha\pm_t\sqrt{\alpha^2-4\gamma}\over 2}\qquad \mbox{(for } \beta=0 \mbox{ only)}.$

Otherwise, continue with

$P = - {\alpha^2 \over 12} - \gamma,$

$Q = - {\alpha^3 \over 108} + {\alpha \gamma \over 3} - {\beta^2 \over 8},$

$R = {Q\over 2} \pm \sqrt{{Q^{2}\over 4}+{P^{3}\over 27}}$ , (either sign of the square root will do)

$U = \sqrt[3]{R}$ , (there are 3 complex roots, any one of them will do)

$y = - {5 \over 6} \alpha -U + \begin{cases}U=0 &\to 0\\U\ne 0 &\to {P\over 3U}\end{cases},$

$W=\sqrt{ \alpha + 2 y}$

$x = - {B \over 4 A} + { \pm_s W \pm_t \sqrt{-\left(3\alpha + 2 y \pm_s {2\beta\over W} \right) }\over 2 }.$

The two ±_s must have the same sign, the ±_t is independent. To get all roots, compute x for ±_s,±_t = +,+ and for +,− and for −,+ and for −,−. This formula handles repeated roots without problem.

Ferrari was the first to discover one of these labyrinthine solutions. In Greek mythology, the Labyrinth ( Greek λαβύρινθος labyrinthos) was an elaborate structure designed and built by the legendary artificer The equation which he solved was

x 4 + 6 x 2 - 60 x + 36 = 0

which was already in depressed form. It has a pair of solutions which can be found with the set of formulas shown above.

Ferrari's solution in the special case of real coefficients

If the coefficients of the quartic equation are real then the nested depressed cubic equation (5) also has real coefficients, thus it has at least one real root.

Furthermore the cubic function

C (v) = v 3 + P v + Q,

where P and Q are given by (5) has the properties that

$C({\alpha \over 3}) = {-\beta^2 \over 8} < 0 \$ and

$\lim_{v\to \infty}{C(v)}=\infty,$ where α and β are given by (1). This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve.

This means that (5) has a real root greater than $\alpha \over 3$ , and therefore that (4) has a real root greater than $-\alpha \over 2$ .

Using this root the term $\sqrt{\alpha + 2 y}$ in (8) is always real, which ensures that the two quadratic equations (8) have real coefficients^[2].

Obtaining alternative solutions the hard way

It could happen that one only obtained one solution through the seven formulae above, because not all four sign patterns are tried for four solutions, and the solution obtained is complex. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted It may also be the case that one is only looking for a real solution. Let x₁ denote the complex solution. If all the original coefficients A, B, C, D and E are real -- which should be the case when one desires only real solutions -- then there is another complex solution x₂ which is the complex conjugate of x₁. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. If the other two roots are denoted as x₃ and x₄ then the quartic equation can be expressed as

(x - x 1)(x - x 2)(x - x 3)(x - x 4) = 0,

but this quartic equation is equivalent to the product of two quadratic equations:

$(x - x_1) (x - x_2) = 0 \qquad \qquad (9)$

and

$(x - x_3) (x - x_4) = 0. \qquad \qquad (10)$

Since

$x_2 = x_1^\star$

then

$\begin{matrix} (x-x_1)(x-x_2)&=&x^2-(x_1+x_1^\star)x+x_1x_1^\star\qquad\qquad\qquad\quad \\ &=&x^2-2\,\mathrm{Re}(x_1)x+[\mathrm{Re}(x_1)]^2+[\mathrm{Im}(x_1)]^2. \end{matrix}$

Let

$a = - 2 \, \mathrm{Re}(x_1),$

$b = \left[ \mathrm{Re}( x_1) \right]^{2} + \left[ \mathrm{Im}(x_1) \right]^{2}$

so that equation (9) becomes

$x^2 + a x + b = 0. \qquad \qquad (11)$

Also let there be (unknown) variables w and v such that equation (10) becomes

$x^2 + w x + v = 0. \qquad \qquad (12)$

Multiplying equations (11) and (12) produces

$x^4 + (a + w) x^3 + (b + w a + v) x^2 + (w b + v a) x + v b = 0. \qquad \qquad (13)$

Comparing equation (13) to the original quartic equation, it can be seen that

$a + w = {B \over A},$

$b + w a + v = {C \over A},$

$w b + v a = {D \over A},$

and

$v b = {E \over A}.$

Therefore

$w = {B \over A} - a = {B \over A} + 2 \mathrm{Re}(x_1),$

$v = {E \over A b} = {E \over A \left( [ \mathrm{Re}(x_1) ]^2 + [ \mathrm{Im}(x_1) ]^2 \right) }.$

Equation (12) can be solved for x yielding

$x_3 = {-w + \sqrt{w^2 - 4 v} \over 2},$

$x_4 = {-w - \sqrt{w^2 - 4 v} \over 2}.$

One of these two solutions should be the desired real solution.

Alternative methods

Quick and memorable solution from first principles

Most textbook solutions of the quartic equation require a magic substitution that is almost impossible to memorize. Here is a way to approach it that makes it easy to understand.

The job is done if we can factorize the quartic equation into a product of two quadratics. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Let

$\begin{array}{lcl} 0 = x^4 + bx^3 + cx^2 + dx + e & = & (x^2 + px + q)(x^2 + rx + s) \\ & = & x^4 + (p + r)x^3 + (q + s + pr)x^2 + (ps + qr)x + qs \end{array}$

By equating coefficients, this results in the following set of simultaneous equations:

$\begin{array}{lcl} b & = & p + r \\ c & = & q + s + pr \\ d & = & ps + qr \\ e & = & qs \end{array}$

This is harder to solve than it looks, but if we start again with a depressed quartic where $b = 0$ , which can be obtained by substituting $(x - b / 4)$ for $x$ , then $r = - p$ , and:

$\begin{array}{lcl} c + p^2 & = & s + q \\ d/p & = & s - q \\ e & = & sq \end{array}$

It's now easy to eliminate both $s$ and $q$ by doing the following:

$\begin{array}{lcl} (c + p^2)^2 - (d/p)^2 & = & (s + q)^2 - (s - q)^2 \\ & = & 4sq \\ & = & 4e \end{array}$

If we set $P = p 2$ , then this equation turns into the cubic equation:

$P^3 + 2cP^2 + (c^2 - 4e)P - d^2 = 0\,$

which is solved elsewhere. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Once you have $p$ , then:

$\begin{array}{lcl} r & = & -p \\ 2s & = & c + p^2 + d/p \\ 2q & = & c + p^2 - d/p \end{array}$

The symmetries in this solution are easy to see. 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 $p$ for the square root of $P$ merely exchanges the two quadratics with one another.

Galois theory and factorization

The symmetric group S₄ on four elements has the Klein four-group as a normal subgroup. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. This suggests using a resolvent whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots. In Mathematics, a Hadamard matrix is a Square matrix whose entries are either +1 or &minus1 and whose rows are mutually Orthogonal. Suppose r_i for i from 0 to 3 are roots of

$x^4 + bx^3 + cx^2 + dx + e = 0\qquad (1)$

If we now set

$\begin{align} s_0 &= \tfrac12(r_0 + r_1 + r_2 + r_3), \\s_1 &= \tfrac12(r_0 - r_1 + r_2 - r_3), \\s_2 &= \tfrac12(r_0 + r_1 - r_2 - r_3), \\s_3 &= \tfrac12(r_0 - r_1 - r_2 + r_3),\end{align}$

then since the transformation is an involution we may express the roots in terms of the four s_i in exactly the same way. Since we know the value s₀ = -b/2, we really only need the values for s₁, s₂ and s₃. These we may find by expanding the polynomial

$(z^2 - s_1^2)(z^2-s_2^2)(z^2-s_3^2)\qquad (2)$

which if we make the simplifying assumption that b=0, is equal to

${z}^{6}+2c\,{z}^{4}+({c}^{2}-4e)\,{z}^{2}-{d}^{2}\qquad(3)$

This polynomial is of degree six, but only of degree three in z², and so the corresponding equation is solvable. By trial we can determine which three roots are the correct ones, and hence find the solutions of the quartic.

We can remove any requirement for trial by using a root of the same resolvent polynomial for factoring; if w is any root of (3), and if $F_1 = {x}^{2}+wx+1/2\,{w}^{2}+1/2\,c-1/2\,{\frac {{c}^{2}w}{d}}-1/2\,{\frac {{w}^{5}}{d}}-{\frac {c{w}^{3}}{d}}+2\,{\frac {ew}{d}}$ $F_2 = {x}^{2}-wx+1/2\,{w}^{2}+1/2\,c+1/2\,{\frac {{w}^{5}}{d}}+{\frac {c{w}^{3}}{d}}-2\,{\frac {ew}{d}}+1/2\,{\frac {{c}^{2}w}{d}}$ then

$F_1 F_2 = x^4 + cx^2 + dx + e\,\,(4)$

We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.

References

^ Stewart, Ian, Galois Theory, Third Edition (Chapman & Hall/CRC Mathematics, 2004)
^ Carstensen, Jens, Komplekse tal, First Edition, (Systime 1981), ISBN: 87-87454-71-8

A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. In Mathematics, a quintic equation is a Polynomial Equation of degree five In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Lodovico Ferrari ( February 2, 1522 – October 5, 1565) was an Italian Mathematician. In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the

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