Editing Quintic function

{{short description|Polynomial function of degree 5}}
[[File:Quintic polynomial.svg|thumb|right|233px|Graph of a polynomial of degree 5, with 3 real zeros (roots) and 4 [[critical point (mathematics)|critical points]] ]]

In [[mathematics]], a '''quintic function''' is a [[function (mathematics)|function]] of the form

:<math>g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,</math>

where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, {{mvar|e}} and {{mvar|f}} are members of a [[field (mathematics)|field]], typically the [[rational number]]s, the [[real number]]s or the [[complex number]]s, and {{mvar|a}} is nonzero. In other words, a quintic function is defined by a [[polynomial]] of [[Degree of a polynomial|degree]] five.

Because they have an odd degree, normal quintic functions appear similar to normal [[cubic function]]s when graphed, except they may possess one additional [[Maxima and minima|local maximum]] and one additional local minimum. The [[derivative]] of a quintic function is a [[quartic function]].

Setting {{math|''g''(''x'') {{=}} 0}} and assuming {{math|''a'' ≠ 0}} produces a '''quintic equation''' of the form:
:<math>ax^5+bx^4+cx^3+dx^2+ex+f=0.\,</math> 
Solving quintic equations in terms of [[Nth_root|radicals]] (''n''th roots) was a major problem in algebra from the 16th century, when [[cubic equation|cubic]] and [[quartic equation]]s were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the [[Abel–Ruffini theorem]].

==Finding roots of a quintic equation==

Finding the [[Zero of a function|roots]] (zeros) of a given polynomial has been a prominent mathematical problem.

Solving [[Linear equation|linear]], [[Quadratic equation|quadratic]], [[Cubic equation|cubic]] and [[quartic equation]]s in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions. However, there is no [[algebraic expression]] (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the [[Abel–Ruffini theorem]], first asserted in 1799 and completely proven in 1824. This result also holds for equations of higher degree. An example of a quintic whose roots cannot be expressed in terms of radicals is {{math| ''x''{{sup|5}} − ''x'' + 1 {{=}} 0}}.

Numerical approximations of quintics roots can be computed with [[Polynomial root-finding algorithms|root-finding algorithms for polynomials]]. Although some quintics may be solved in terms of radicals, the solution is generally too complicated to be used in practice.

==Solvable quintics==

Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is [[irreducible polynomial|reducible]], such as {{math|''x''<sup>5</sup> − ''x''<sup>4</sup> − ''x'' + 1 {{=}} (''x''<sup>2</sup> + 1)(''x'' + 1)(''x'' − 1)<sup>2</sup>}}. For example, it has been shown<ref>{{cite journal |last1=Elia |first1=M. |last2=Filipponi |first2=P. |title=Equations of the Bring–Jerrard Form, the Golden Section, and Square Fibonacci Numbers |journal=The Fibonacci Quarterly |date=1998 |volume=36 |issue=3 |pages=282–286 |url=https://www.fq.math.ca/Scanned/36-3/elia.pdf}}</ref> that

:<math>x^5-x-r=0</math>

has solutions in radicals [[if and only if]] it has an [[integer]] solution or ''r'' is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible.

As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only irreducible quintic equations are considered in the remainder of this section, and the term "quintic" will refer only to irreducible quintics. A '''solvable quintic''' is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals.

To characterize solvable quintics, and more generally solvable polynomials of higher degree, [[Évariste Galois]] developed techniques which gave rise to [[group theory]] and [[Galois theory]]. Applying these techniques, [[Arthur Cayley]] found a general criterion for determining whether any given quintic is solvable.<ref>A. Cayley, "On a new auxiliary equation in the theory of equation of the fifth order", ''Philosophical Transactions of the Royal Society of London'' '''151''':263-276 (1861) {{doi|10.1098/rstl.1861.0014}}</ref> This criterion is the following.<ref>This formulation of Cayley's result is extracted from Lazard (2004) paper.</ref>

Given the equation
:<math> ax^5+bx^4+cx^3+dx^2+ex+f=0,</math>
the [[Tschirnhaus transformation]] {{math|''x'' {{=}} ''y'' − {{sfrac|''b''|5''a''}}}}, which depresses the quintic (that is, removes the term of degree four), gives the equation

:<math> y^5+p y^3+q y^2+r y+s=0,</math>

where

:<math>\begin{align}p &= \frac{5ac-2b^2}{5a^2}\\[4pt]
q &= \frac{25a^2d-15abc+4b^3}{25a^3}\\[4pt]
r &= \frac{125a^3e-50a^2bd+15ab^2c-3b^4}{125a^4}\\[4pt]
s &= \frac{3125 a^4f-625a^3 be+125a^2b^2 d-25ab^3 c+4 b^5}{3125a^5}\end{align}</math>

Both quintics are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial {{math|''P''<sup>2</sup> − 1024 ''z'' Δ}}, named '''{{vanchor|Cayley's resolvent}}''', has a rational root in {{mvar|z}}, where

:<math>\begin{align} 
P = {} &z^3-z^2(20r+3p^2)- z(8p^2r - 16pq^2- 240r^2 + 400sq - 3p^4)\\[4pt]
&- p^6 + 28p^4r- 16p^3q^2- 176p^2r^2- 80p^2sq + 224prq^2- 64q^4 \\[4pt]
&+ 4000ps^2 + 320r^3- 1600rsq
\end{align}
</math>

and

:<math>\begin{align}
\Delta = {} &-128p^2r^4+3125s^4-72p^4qrs+560p^2qr^2s+16p^4r^3+256r^5+108p^5s^2 \\[4pt]
&-1600qr^3s+144pq^2r^3-900p^3rs^2+2000pr^2s^2-3750pqs^3+825p^2q^2s^2 \\[4pt]
&+2250q^2rs^2+108q^5s-27q^4r^2-630pq^3rs+16p^3q^3s-4p^3q^2r^2. \end{align}
</math>
Cayley's result allows us to test if a quintic is solvable. If it is the case, finding its roots is a more difficult problem, which consists of expressing the roots in terms of radicals involving the coefficients of the quintic and the rational root of Cayley's resolvent.

In 1888, [[George Paxton Young]] described how to solve a solvable quintic equation, without providing an explicit formula;<ref>George Paxton Young, "Solvable Quintic Equations with Commensurable Coefficients", ''American Journal of Mathematics'' '''10''':99–130 (1888), {{JSTOR|2369502}}</ref> in 2004, [[Daniel Lazard]] wrote out a three-page formula.<ref>{{harvtxt|Lazard|2004|p=207}}</ref>

===Quintics in Bring–Jerrard form===

There are several parametric representations of solvable quintics of the form {{math|''x''<sup>5</sup> + ''ax'' + ''b'' {{=}} 0}}, called the [[Bring–Jerrard form]].

During the second half of the 19th century, John Stuart Glashan, George Paxton Young, and [[Carl Runge]] gave such a parameterization: an [[irreducible polynomial|irreducible]] quintic with rational coefficients in Bring–Jerrard form
is solvable if and only if either {{math|''a'' {{=}} 0}} or it may be written
:<math>x^5 + \frac{5\mu^4(4\nu + 3)}{\nu^2 + 1}x + \frac{4\mu^5(2\nu + 1)(4\nu + 3)}{\nu^2 + 1} = 0</math>
where {{math|''μ''}} and {{math|''ν''}} are rational.

In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,
:<math>x^5 + \frac{5e^4( 4c + 3)}{c^2 + 1}x + \frac{-4e^5(2c-11)}{c^2 + 1} = 0.</math>

The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression
:<math>b = \frac{4}{5} \left(a+20 \pm 2\sqrt{(20-a)(5+a)}\right)</math>
where {{tmath|1=a=5\tfrac{4\nu+3}{\nu^2+1} }}. Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second.

The substitution {{tmath|1=c = - \tfrac{m}{\ell^5}, }} {{tmath|1=e = \tfrac{1}{\ell} }} in the Spearman–Williams parameterization allows one to not exclude the special case {{math|''a'' {{=}} 0}}, giving the following result:

If {{mvar|a}} and {{mvar|b}} are rational numbers, the equation {{math|''x''<sup>5</sup> + ''ax'' + ''b'' {{=}} 0}} is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ''ℓ'' and {{mvar|m}} such that
:<math>a=\frac{5 \ell (3 \ell^5-4 m)}{m^2+\ell^{10}}\qquad b=\frac{4(11 \ell^5+2 m)}{m^2+\ell^{10}}.</math>

===Roots of a solvable quintic===
A polynomial equation is solvable by radicals if its [[Galois group]] is a [[solvable group]]. In the case of irreducible quintics, the Galois group is a subgroup of the [[symmetric group]] {{math|''S''<sub>5</sub>}} of all permutations of a five element set, which is solvable if and only if it is a subgroup of the group {{math|''F''<sub>5</sub>}}, of order {{math|20}}, generated by the cyclic permutations {{math|(1 2 3 4 5)}} and {{math|(1 2 4 3)}}.

If the quintic is solvable, one of the solutions may be represented by an [[algebraic expression]] involving a fifth root and at most two square roots, generally [[nested radical|nested]]. The other solutions may then be obtained either by changing the fifth root or by multiplying all the occurrences of the fifth root by the same power of a [[root of unity|primitive 5th root of unity]], such as
:<math>\frac{\sqrt{-10-2\sqrt{5}}+\sqrt{5}-1}{4}.</math>

In fact, all four primitive fifth roots of unity may be obtained by changing the signs of the square roots appropriately; namely, the expression
:<math>\frac{\alpha\sqrt{-10-2\beta\sqrt{5}}+\beta\sqrt{5}-1}{4},</math>
where <math> \alpha, \beta \in \{-1,1\}</math>, yields the four distinct primitive fifth roots of unity.

It follows that one may need four different square roots for writing all the roots of a solvable quintic. Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation {{math|''x''<sup>5</sup> − 5''x''<sup>4</sup> + 30''x''<sup>3</sup> − 50''x''<sup>2</sup> + 55''x'' − 21 {{=}} 0}}, for which the only real solution is

: <math>x=1+\sqrt[5]{2}-\left(\sqrt[5]{2}\right)^2+\left(\sqrt[5]{2}\right)^3-\left(\sqrt[5]{2}\right)^4.</math>

An example of a more complicated (although small enough to be written here) solution is the unique real root of {{math|''x''<sup>5</sup> − 5''x'' + 12 {{=}} 0}}. Let {{math|''a'' {{=}} {{sqrt|2''φ''<sup>−1</sup>}}}}, {{math|''b'' {{=}} {{sqrt|2''φ''}}}}, and {{math|''c'' {{=}} {{radic|5|4}}}}, where {{math|''φ'' {{=}} {{sfrac|1+{{sqrt|5}}|2}}}} is the [[golden ratio]]. Then the only real solution {{math|''x'' {{=}} −1.84208...}} is given by

: <math>-cx = \sqrt[5]{(a+c)^2(b-c)} + \sqrt[5]{(-a+c)(b-c)^2} + \sqrt[5]{(a+c)(b+c)^2} - \sqrt[5]{(-a+c)^2(b+c)} \,,</math>

or, equivalently, by

:<math>x = \sqrt[5]{y_1}+\sqrt[5]{y_2}+\sqrt[5]{y_3}+\sqrt[5]{y_4}\,,</math>

where the {{math|''y<sub>i</sub>''}} are the four roots of the [[quartic equation]]

:<math>y^4+4y^3+\frac{4}{5}y^2-\frac{8}{5^3}y-\frac{1}{5^5}=0\,.</math>

More generally, if an equation {{math|1=''P''(''x'') = 0}} of prime degree {{math|''p''}} with rational coefficients is solvable in radicals, then one can define an auxiliary equation {{math|1=''Q''(''y'') = 0}} of degree {{math|''p'' − 1}}, also with rational coefficients, such that each root of {{math|''P''}} is the sum of {{math|''p''}}-th roots of the roots of {{math|''Q''}}. These {{math|''p''}}-th roots were introduced by [[Joseph-Louis Lagrange]], and their products by {{math|''p''}} are commonly called [[Lagrange resolvent]]s. The computation of {{math|''Q''}} and its roots can be used to solve {{math|1=''P''(''x'') = 0}}. However these {{math|''p''}}-th roots may not be computed independently (this would provide {{math|''p''<sup>''p''−1</sup>}} roots instead of {{math|''p''}}). Thus a correct solution needs to express all these {{math|''p''}}-roots in term of one of them. Galois theory shows that this is always theoretically possible, even if the resulting formula may be too large to be of any use.

It is possible that some of the roots of {{math|''Q''}} are rational (as in the first example of this section) or some are zero. In these cases, the formula for the roots is much simpler, as for the solvable [[de Moivre]] quintic{{anchor|de Moivre quintic}}

:<math>x^5+5ax^3+5a^2x+b = 0\,,</math>

where the auxiliary equation has two zero roots and reduces, by factoring them out, to the [[quadratic equation]]

:<math>y^2+by-a^5 = 0\,,</math>

such that the five roots of the de Moivre quintic are given by

:<math>x_k = \omega^k\sqrt[5]{y_i} -\frac{a}{\omega^k\sqrt[5]{y_i}},</math>

where ''y<sub>i</sub>'' is any root of the auxiliary quadratic equation and ''ω'' is any of the four [[primitive root of unity|primitive 5th roots of unity]].  This can be easily generalized to construct a solvable [[septic equation|septic]] and other odd degrees, not necessarily prime.

===Other solvable quintics===

There are infinitely many solvable quintics in Bring–Jerrard form which have been parameterized in a preceding section.

Up to the scaling of the variable, there are exactly five solvable quintics of the shape <math>x^5+ax^2+b</math>, which are<ref>{{cite web |first=Noam |last=Elkies |title=Trinomials {{nobr|a x{{sup|n}} + b x + c}} with interesting Galois groups |url=https://www.math.harvard.edu/~elkies/trinomial.html |publisher=[[Harvard University]]}}</ref> (where ''s'' is a scaling factor):
:<math>x^5-2s^3x^2-\frac{s^5}{5} </math>
:<math> x^5-100s^3x^2-1000s^5</math>
:<math>x^5-5s^3x^2-3s^5 </math>
:<math>x^5-5s^3x^2+15s^5 </math>
:<math> x^5-25s^3x^2-300s^5</math>

Paxton Young (1888) gave a number of examples of solvable quintics:
:{| <math>x^5+3x^2+2x-1 </math> ||
|-
| <math> x^5-10x^3-20x^2-1505x-7412</math> ||
|-
| <math>x^5+\frac{625}{4}x+3750 </math> ||
|-
| <math>x^5-\frac{22}{5}x^3-\frac{11}{25}x^2+\frac{462}{125}x+\frac{979}{3125} </math> ||
|-
| <math>x^5+20x^3+20x^2+30x+10 </math> || <math>~\qquad ~</math> Root: <math> \sqrt[5]{2}-\sqrt[5]{2}^2+\sqrt[5]{2}^3-\sqrt[5]{2}^4</math>
|-
|<math>x^5-20x^3+250x-400 </math> ||
|-
| <math>x^5-5x^3+\frac{85}{8}x-\frac{13}{2} </math> ||
|-
|<math> x^5+\frac{20}{17}x+\frac{21}{17}</math> ||
|-
|<math>x^5-\frac{4}{13}x+\frac{29}{65}</math> ||
|-
|<math> x^5+\frac{10}{13}x+\frac{3}{13}	</math> ||
|-
| <math> x^5+110(5x^3+60x^2+800x+8320)</math> ||
|-
| <math>x^5-20 x^3 -80x^2 -150x -656</math>||
|-
| <math> x^5 -40x^3 +160x^2 +1000x -5888</math> ||
|-
|<math> x^5-50x^3-600x^2-2000x-11200</math> ||
|-
| <math>x^5+110(5x^3 + 20x^2 -360x +800)</math> ||
|-
| <math> x^5 -20x^3 +170x + 208</math> ||
|}

An infinite sequence of solvable quintics may be constructed, whose roots are sums of {{mvar|n}}th [[roots of unity]], with {{nobr|{{math|''n'' {{=}} 10''k'' + 1}}}} being a [[prime number]]:

:{|
|-
| <math>x^5+x^4-4x^3-3x^2+3x+1</math> || || Roots: <math>2\cos\left(\frac{2k\pi}{11}\right)</math>
|-
| <math> x^5+x^4-12x^3-21x^2+x+5</math> || || Root: <math> \sum_{k=0}^5 e^\frac{2i\pi 6^k }{31}</math>
|-
| <math>x^5+x^4-16x^3+5x^2+21x-9</math> || || Root: <math>\sum_{k=0}^7 e^\frac{2i\pi 3^k }{41}</math>
|-
| <math>x^5+x^4-24x^3-17x^2+41x-13</math> || <math>~\qquad ~</math> || {{nowrap|1= Root: <math>\sum_{k=0}^{11} e^\frac{2i\pi (21)^k }{61}</math>}}
|-
| <math>x^5+x^4 - 28x^3 + 37x^2 + 25x + 1</math> || || {{nowrap|1= Root: <math>\sum_{k=0}^{13} e^\frac{2i\pi (23)^k }{71}</math>}}
|}

There are also two parameterized families of solvable quintics:
The Kondo–Brumer quintic,

:<math> x^5 + (a-3)\,x^4 + (-a+b+3)\,x^3 + (a^2-a-1-2b)\,x^2 + b\,x + a = 0 </math>

and the family depending on the parameters <math>a, \ell, m</math>
:<math>	x^5 - 5\,p \left( 2\,x^3 + a\,x^2 + b\,x \right) - p\,c = 0 </math>

where

::<math> p = \tfrac{1}{4} \left[\, \ell^2 (4m^2 + a^2) - m^2 \,\right] \;,</math>
:
::<math> b = \ell \, ( 4m^2 + a^2 ) - 5p - 2m^2 \;,</math>
:
::<math> c = \tfrac{1}{2} \left[\, b(a + 4m) - p(a - 4m) - a^2m  \,\right] \;.</math>

===''Casus irreducibilis''===

Analogously to [[cubic equation]]s, there are solvable quintics which have five real roots all of whose solutions in radicals involve roots of complex numbers. This is ''[[casus irreducibilis]]'' for the quintic, which is discussed in Dummit.<ref>David S. Dummit [http://www.emba.uvm.edu/~dummit/quintics/solvable.pdf Solving Solvable Quintics]</ref>{{rp|p.17}} Indeed, if an irreducible quintic has all roots real, no root can be expressed purely in terms of real radicals (as is true for all polynomial degrees that are not powers of 2).

==Beyond radicals==

About 1835, [[George Jerrard|Jerrard]] demonstrated that quintics can be solved by using [[ultraradical]]s (also known as Bring radicals), the unique real root of {{math|''t''<sup>5</sup> + ''t'' − ''a'' {{=}} 0}} for real numbers {{math|''a''}}. In 1858, [[Charles Hermite]] showed that the Bring radical could be characterized in terms of the Jacobi [[theta function]]s and their associated [[elliptic modular function]]s, using an approach similar to the more familiar approach of solving [[cubic equation]]s by means of [[trigonometric function]]s.  At around the same time, [[Leopold Kronecker]], using [[group theory]], developed a simpler way of deriving Hermite's result, as had [[Francesco Brioschi]].  Later, [[Felix Klein]] came up with a method that relates the symmetries of the [[icosahedron]], [[Galois theory]], and the elliptic modular functions that are featured in Hermite's solution, giving an explanation for why they should appear at all, and developed his own solution in terms of [[generalized hypergeometric function]]s.<ref>{{Harv|Klein|1888}}; a modern exposition is given in {{Harv|Tóth|2002|loc=Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, [https://books.google.com/books?id=i76mmyvDHYUC&pg=PA66 p. 66]}}</ref> Similar phenomena occur in degree {{math|7}} ([[septic equation]]s) and {{math|11}}, as studied by Klein and discussed in {{slink|Icosahedral symmetry|Related geometries}}.

===Solving with Bring radicals===
{{Main article|Bring radical}}

A [[Tschirnhaus transformation]], which may be computed by solving a [[quartic equation]], reduces the general quintic equation of the form 
:<math>x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\,</math> 
to the [[Bring–Jerrard normal form]] {{math|''x''<sup>5</sup> − ''x'' + ''t'' {{=}} 0}}.

The roots of this equation cannot be expressed by radicals. However, in 1858, [[Charles Hermite]] published the first known solution of this equation in terms of [[elliptic function]]s.<ref name="hermite">{{cite journal
 | last = Hermite
 | first = Charles
 | year = 1858
 | title =  Sur la résolution de l'équation du cinquième degré 
 | journal = Comptes Rendus de l'Académie des Sciences
 | volume = XLVI
 | issue = I
 | pages = 508–515}}</ref>
At around the same time [[Francesco Brioschi]]<ref>
{{cite journal
 | last = Brioschi
 | first = Francesco
 | year = 1858
 | title =  Sul Metodo di Kronecker per la Risoluzione delle Equazioni di Quinto Grado 
 | journal = Atti Dell'i. R. Istituto Lombardo di Scienze, Lettere ed Arti
 | volume = I
 | pages = 275–282}}</ref> 
and [[Leopold Kronecker]]<ref>
{{cite journal
 | last = Kronecker
 | first = Leopold
 | year = 1858
 | title =  Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite 
 | journal = Comptes Rendus de l'Académie des Sciences
 | volume = XLVI
 | issue = I
 | pages = 1150–1152}}</ref>
came upon equivalent solutions.

See [[Bring radical]] for details on these solutions and some related ones.

==Application to celestial mechanics==
Solving for the locations of the [[Lagrangian point]]s of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic.

More precisely, the locations of ''L''<sub>2</sub> and ''L''<sub>1</sub> are the solutions to the following equations, where the gravitational forces of two masses on a third (for example, Sun and Earth on satellites such as [[Gaia probe|Gaia]] and the [[James Webb Space Telescope]] at ''L''<sub>2</sub> and [[Solar and Heliospheric Observatory|SOHO]] at ''L''<sub>1</sub>) provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun:

: <math>\frac{G m M_S}{(R \pm r)^2} \pm \frac{G m M_E}{r^2} = m \omega^2 (R \pm r)</math>

The ± sign corresponds to ''L''<sub>2</sub> and ''L''<sub>1</sub>, respectively; ''G'' is the [[gravitational constant]], ''ω'' the [[angular velocity]], ''r'' the distance of the satellite to Earth, ''R'' the distance Sun to Earth (that is, the [[semi-major axis]] of Earth's orbit), and ''m'', ''M<sub>E</sub>'', and ''M<sub>S</sub>'' are the respective masses of satellite, [[Earth]], and [[Sun]].

Using Kepler's Third Law <math>\omega^2=\frac{4 \pi^2}{P^2}=\frac{G (M_S+M_E)}{R^3}</math> and rearranging all terms yields the quintic

: <math>a r^5 + b r^4 + c r^3 + d r^2 + e r + f = 0</math>

with:
:<math> \begin{align}
&a = \pm (M_S + M_E),\\
&b = + (M_S + M_E) 3 R,\\
&c = \pm (M_S + M_E) 3 R^2,\\
&d = + (M_E \mp M_E) R^3\ (\text{thus } d = 0\text{ for } L_2),\\
&e = \pm M_E 2 R^4,\\
&f = \mp M_E R^5.
\end{align}</math>

Solving these two quintics yields {{math|1=''r'' = 1.501 × 10<sup>9</sup> ''m''}} for ''L''<sub>2</sub> and {{math|1=''r'' = 1.491 × 10<sup>9</sup> ''m''}} for ''L''<sub>1</sub>. The [[List of objects at Lagrangian points|Sun–Earth Lagrangian points]] ''L''<sub>2</sub> and ''L''<sub>1</sub> are usually given as 1.5 million km from Earth.

If the mass of the smaller object (''M''<sub>E</sub>) is much smaller than the mass of the larger object (''M''<sub>S</sub>), then the quintic equation can be greatly reduced and L<sub>1</sub> and L<sub>2</sub> are at approximately the radius of the [[Hill sphere]], given by:

: <math>r \approx R \sqrt[3]{\frac{M_E}{3 M_S}}</math>

That also yields {{math|1=''r'' = 1.5 × 10<sup>9</sup> ''m''}} for satellites at L<sub>1</sub> and L<sub>2</sub> in the Sun-Earth system.

==See also==
* [[Sextic equation]]
* [[Septic function]]
* [[Theory of equations]]
* [[Principal equation form]]

==Notes==
{{reflist}}

==References==
* Charles Hermite, "Sur la résolution de l'équation du cinquème degré", ''Œuvres de Charles Hermite'', '''2''':5–21, Gauthier-Villars, 1908.
* {{cite book |first=Felix |last=Klein |url=https://archive.org/details/cu31924059413439 |title=Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree |translator-first=George Gavin |translator-last=Morrice |publisher=Trübner & Co. |date=1888 |isbn=0-486-49528-0}}
* Leopold Kronecker, "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite", ''Comptes Rendus de l'Académie des Sciences'', '''46''':1:1150–1152 1858.
* Blair Spearman and Kenneth S. Williams, "Characterization of solvable quintics {{math|''x''<sup>5</sup> + ''ax'' + ''b''}}, ''American Mathematical Monthly'', '''101''':986–992 (1994).
* Ian Stewart, ''Galois Theory'' 2nd Edition, Chapman and Hall, 1989. {{isbn|0-412-34550-1}}. Discusses Galois Theory in general including a proof of insolvability of the general quintic.
* [[Jörg Bewersdorff]], ''Galois theory for beginners: A historical perspective'', American Mathematical Society, 2006. {{isbn|0-8218-3817-2}}. Chapter 8 ({{webarchive|url=https://web.archive.org/web/20100331181637/http://www.ams.org/bookstore/pspdf/stml-35-prev.pdf|title=The solution of equations of the fifth degree|date=31 March 2010}}) gives a description of the solution of solvable quintics {{math|''x''<sup>5</sup> + ''cx'' + ''d''}}.
* Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of Tschirnhaus, Bring and Jerrard," ''ACM SIGSAM Bulletin'', Vol. 37, No. 3, September 2003, pp.&nbsp;90–94.
* Ehrenfried Walter von Tschirnhaus, "A method for removing all intermediate terms from a given equation," ''ACM SIGSAM Bulletin'', Vol. 37, No. 1, March 2003, pp.&nbsp;1–3.
* {{Cite book | last1 = Lazard | first1 = Daniel | chapter = Solving quintics in radicals | title = The Legacy of Niels Henrik Abel | editor1 = [[Olav Arnfinn Laudal]] | editor2 = [[Ragni Piene]] | location = Berlin | pages = 207&ndash;225 | year = 2004 | isbn = 3-540-43826-2 | url = https://www.loria.fr/publications/2002/A02-R-449/A02-R-449.ps | archive-url = https://web.archive.org/web/20050106213419/http://www.loria.fr/publications/2002/A02-R-449/A02-R-449.ps | archive-date=January 6, 2005 }}
* {{citation | title = Finite Möbius groups, minimal immersions of spheres, and moduli| first = Gábor | last = Tóth | year = 2002 }}

==External links==
* [https://mathworld.wolfram.com/QuinticEquation.html Mathworld - Quintic Equation] – more details on methods for solving Quintics.
* [http://www.emba.uvm.edu/~dummit/quintics/solvable.pdf Solving Solvable Quintics] – a method for solving solvable quintics due to David S. Dummit.
* [https://web.archive.org/web/20090226035640/http://www.sigsam.org/bulletin/articles/143/tschirnhaus.pdf A method for removing all intermediate terms from a given equation] - a recent English translation of Tschirnhaus' 1683 paper.
* [https://www.ams.org/journals/notices/202404/noti2923/noti2923.html Bruce Bartlett:''The Quintic, the Icosahedron, and Elliptic Curves''], [[AMS Notices]] (April 2024)

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[[Category:Equations]]
[[Category:Galois theory]]
[[Category:Polynomial functions]]