Quartic equation

In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is

:$ax^4+bx^3+cx^2+dx+e=0 \,$
where ''a'' â‰  0.

The quartic is the highest order polynomial equation that can be solved by radicals in the general case.

History

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this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately. 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 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 notes left by Évariste Galois before his death in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result.

Special case solutions

Consider a quartic equation expressed in the form $a_0x^4+a_1x^3+a_2x^2+a_3x+a_4=0$:

There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is non-zero. However, since the general method is quite complex and susceptible to errors in execution, it is better to apply one of the special cases listed below if possible.

Degenerate case

If the constant term ''a''₄ = 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 −

Call our quartic polynomial . Since 1 raised to any power is 1,
: $Q(1) = a_0 + a_1 + a_2 + a_3 + a_4 \ .$
Thus if $\ a_0 + a_1 + a_2 + a_3 + a_4 = 0\ ,$ and so = 1 is a root of . It can similarly be shown that if $\ a_0 + a_2 + a_4 = a_1 + a_3 \ ,$ = −1 is a root.

In either case the full quartic can then be divided by the factor or respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots.

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

Alternatively, if $\ a_1 = a_0 k\ ,$ $\ a_3 = a_2 k\ ,$ and $\ a_4 = 0\ ,$ then and become two known roots. divided by is a quadratic polynomial.

Biquadratic equations

A quartic equation where ''a''₃ and ''a''₁ 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 becomes

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

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

:$z=\frac \,\!$

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

:$x_1=+\sqrt\,\!$
:$x_2=-\sqrt\,\!$
:$x_3=+\sqrt\,\!$
:$x_4=-\sqrt\,\!$

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

Quasi-symmetric equations

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

Steps:

# Divide by ''x'' ².
# Use variable change ''z'' = ''x'' + ''m''/''x''.
# So, ''z'' ² = ''x'' ² + (''m''/''x'') ² + 2''m''.

This leads to:

:$a_0(x^2+m^2/x^2) + a_1(x+m/x) + a_2=0$,
:$a_0(z^2-2m) + a_1(z) + a_2=0$,
:$z^2 + (a_1/a_0)z + (a_2/a_0-2m)=0$ (a quadratic in ''z'' = ''x'' + ''m''/''x'')

Multiple roots

If the quartic has a double root, it can be found by taking the polynomial greatest common divisor with its derivative. Then they can be divided out and the resulting quadratic equation solved.

In general, there exist only four possible cases of quartic equations with multiple roots, which are listed below:
# Multiplicity-4 (M4): when the general quartic equation can be expressed as $a(x-l)^4=0$, for some real number $l$. This case can always be reduced to a biquadratic equation.
# Multiplicity-3 (M3): when the general quartic equation can be expressed as $a(x-l)^3 (x-m)=0$, where $l$ and $m$ are two different real numbers. This is the only case that can never be reduced to a biquadratic equation.
# Double Multiplicity-2 (DM2): when the general quartic equation can be expressed as $a(x-l)^2 (x-m)^2=0$, where $l$ and $m$ are two different real numbers or a pair of non-real complex conjugate numbers. This case can also always be reduced to a biquadratic equation.
# Single Multiplicity-2 (SM2): when the general quartic equation can be expressed as $a(x-l)^2 (x-m)(x-n)=0$, where $l$, $m$, and $n$ are three different real numbers or $l$ is a real number and $m$ and $n$ are a pair of non-real complex conjugate numbers. This case is divided into two subcases, those that can be reduced to a biquadratic equation and those that can't.

Consider the case in which the three non-monic coefficients of the depressed quartic equation, $x^4 + p x^2 + q x + r = 0$, can be expressed in terms of the five coefficients of the general quartic equation as follows:
:$p=\frac$
:$q=\frac$
:$r=\frac$,
Then, the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are shown below.
* M4. The general quartic equation corresponds to this case whenever $p=q=r=0$, so the four roots of this equation are given as follows:
:: $x_1=x_2=x_3=x_4=-\frac$.
* M3. The general quartic equation corresponds to this case whenever $p^2=-12r>0$ and $27q^2=-8p^3>0$, so the four roots of this equation are given as follows if $q>0$:
:: $x_1=x_2=x_3=\sqrt-\frac$
:: $x_4=-\sqrt-\frac$
: Otherwise, if $q\le0$:
:: $x_1=x_2=x_3=-\sqrt-\frac$
:: $x_4=\sqrt-\frac$.
* DM2. The general quartic equation corresponds to this case whenever $p^2=4r>0=q$, so the four roots of this equation are given as follows:
:: $x_1=x_3=\sqrt-\frac$ 
:: $x_2=x_4=-\sqrt-\frac$.
* Biquadratic SM2. The general quartic equation corresponds to this subcase of the SM2 equations whenever $p\neq q=r=0$, so the four roots of this equation are given as follows:
:: $x_1=x_2=-\frac$
:: $x_3=\sqrt-\frac$ 
:: $x_4=-\sqrt-\frac$.
* Non-Biquadratic SM2. The general quartic equation corresponds to this subcase of the SM2 equations whenever (p^2+12r)^3= (p^2-36r)+\frac q^2 2>0\neq , so the four roots of this equation are given by the following formula:
:: x=\frac \left \xi\sqrt\pm\sqrt \right \frac,
: where:
:: $s_1=\frac>0$
:: $s_2=-\frac\neq 0$
:: $\xi = \pm 1$.

The general case

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

Converting to a depressed quartic

Let

be the general quartic equation which it is desired to solve. Divide both sides by ,
:$\ x^4 + x^3 + x^2 + x + = 0 \ .$

The first step, if is not already zero, should be to eliminate the ³ term. To do this, change variables from to , such that
:$\ x = u - \ .$
Then
:$\ \left( u - \right)^4 + \left( u - \right)^3 + \left( u - \right)^2 + \left( u - \right) + = 0 \ .$
Expanding the powers of the binomials produces
:$\ \left( u^4 - u^3 + - + \right) + \left( u^3 - + - \right) + \left( u^2 - + \right) + \left( u - \right) + = 0 \ .$
Collecting the same powers of ''u'' yields
:$\ u^4 + \left( + \right) u^2 + \left( - + \right) u + \left( + - + \right) = 0 \ .$

Now rename the coefficients of . Let
:$\begin a & = + \ ,\\ b & = - + \ ,\\ c & = + - + \ . \end$
The resulting equation is

which is a depressed quartic equation.

If $\ b = 0 \$ then we have the special case of a biquadratic equation, which is easily solved, as explained above. Note that the general solution, given below, will ''not'' work for the special case $\ b = 0 \ .$ The equation must be solved as a biquadratic.

In either case, once the depressed quartic is solved for , substituting those values into
:$\ x = u - \$
produces the values for that solve the original quartic.

Solving a depressed quartic when ≠ 0

After converting to a depressed quartic equation
:$u^4 + a u^2 + b u + c = 0$
and excluding the special case = 0, which is solved as a biquadratic, we assume from here on that

We will separate the terms left and right as
:$u^4 = - a u^2 - b u - c$
and add in terms to both sides which make them both into perfect squares.

Let be any solution of this cubic equation:
:$2 y^3 - a y^2 - 2 c y + ( a c - \tfrac14 b^2 ) = ( 2 y - a ) ( y^2 - c ) - \tfrac14 b^2 = 0 \ .$
Then (since ≠ 0)
:$2 y - a \neq 0$
so we may divide by it, giving
:$y^2 - c = \frac \ .$

Then
:$( u^2 + y )^2 = u^4 + 2 y u^2 + y^2 = ( 2 y - a ) u^2 - b u + ( y^2 - c ) = ( 2 y - a ) u^2 - b u + \frac = \left( \sqrt \, u - \frac \right)^2 \ .$

Subtracting, we get the difference of two squares which is the product of the sum and difference of their roots
:$( u^2 + y )^2 - \left( \sqrt \, u - \frac \right)^2 = \left( u^2 + y + \sqrt \, u - \frac \right) \left( u^2 + y - \sqrt \, u + \frac \right) = 0$
which can be solved by applying the quadratic formula to each of the two factors. So the possible values of are:
:$u = \tfrac12 \left( - \sqrt + \sqrt \right) \ ,$
:$u = \tfrac12 \left( - \sqrt - \sqrt \right) \ ,$
:$u = \tfrac12 \left( \sqrt + \sqrt \right) \ ,$ or
:$u = \tfrac12 \left( \sqrt - \sqrt \right) \ .$

Using another from among the three roots of the cubic simply causes these same four values of to appear in a different order. The solutions of the cubic are:
:$\ y = \frac + w - \frac\$
:\ w = \sqrt
using any one of the three possible cube roots. A wise strategy is to choose the sign of the square-root that makes the absolute value of as large as possible.
:$\ p = - \frac - c \ ,$
:$\ q = - \frac + \frac - \frac \ .$

Ferrari's solution

Otherwise, the depressed quartic can be solved by means of a method discovered by Lodovico Ferrari. Once the depressed quartic has been obtained, the next step is to add the valid identity
:$\left(u^2 + a\right)^2 - u^4 - 2 a u^2 = a^2$
to equation (), yielding

The effect has been to fold up the ''u''⁴ term into a perfect square: (''u''² + a)². The second term, ''au''² 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 (), and a corresponding 2''y'' into the coefficient of ''u''² in the right side. To accomplish these insertions, the following valid formulas will be added to equation (),
:$\begin (u^2+a+y)^2-(u^2+a)^2 & = 2y(u^2+a)+ y^2\ \ \\ & = 2yu^2+2ya+y^2, \end$
and
:$0 = (a + 2 y) u^2 - 2 y u^2 - a u^2\,$
These two formulas, added together, produce
:$\left(u^2 + a + y\right)^2 - \left(u^2 + a\right)^2 = \left(a + 2 y\right) u^2 - a u^2 + 2 y a + y^2 \qquad \qquad (y\hbox)\,$
which added to equation () produces
:$\left(u^2 + a + y\right)^2 + b u + c = \left(a + 2 y\right) u^2 + \left(2 y a + y^2 + a^2\right).\,$
This is equivalent to

The objective now is to choose a value for ''y'' such that the right side of equation () 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:
:$\left(s u + t\right)^2 = \left(s^2\right) u^2 + \left(2 s t\right) u + \left(t^2\right).\,$
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:
:$\left(2 s t\right)^2 - 4 \left(s^2\right) \left(t^2\right) = 0.\,$

Therefore to make the right side of equation () into a perfect square, the following equation must be solved:
:$(-b)^2 - 4 \left(2 y + a\right) \left(y^2 + 2 y a + a^2 - c\right) = 0.\,$
Multiply the binomial with the polynomial,
:$b^2 - 4 \left(2 y^3 + 5 a y^2 + \left(4 a^2 - 2 c\right) y + \left(a^3 - a c\right)\right) = 0\,$
Divide both sides by −4, and move the −''b''²/4 to the right,
:$2 y^3 + 5 a y^2 + \left( 4 a^2 - 2 c \right) y + \left( a^3 - a c - \frac \right) = 0$
Divide both sides by 2,

This is a cubic equation in ''y''. Solve for ''y'' using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.

Folding the second perfect square

With the value for ''y'' so selected, it is now known that the right side of equation () is a perfect square of the form
:$\left(s^2\right)u^2 + (2st)u + \left(t^2\right) = \left(\left(\sqrt\right)u + \right)^2$
::(This is correct for both signs of square root, as long as the same sign is taken for both square roots. A ± is redundant, as it would be absorbed by another ± a few equations further down this page.)
so that it can be folded:
:$(a + 2 y) u^2 + (- b) u + \left(y^2 + 2 y a + a^2 - c \right) = \left( \left(\sqrt\right)u + \right)^2.$
::Note: If ''b'' ≠ 0 then ''a'' + 2''y'' ≠ 0. If ''b'' = 0 then this would be a biquadratic equation, which we solved earlier.
Therefore equation () becomes

Equation () 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:

Collecting like powers of ''u'' produces

::Note: The subscript ''s'' of $\pm_s$ and $\mp_s$ is to note that they are dependent.
Equation () is a quadratic equation for ''u''. Its solution is
:$u=\frac.$
Simplifying, one gets
:$u=.$
This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are

::Remember: The two $\pm_s$ come from the same place in equation (), 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:

:$a = - + ,$

:$b = - + ,$

:$c = - + - + .$

If $\,b=0,$ then

::$x=-\pm_s\sqrt\qquad \mbox b=0 \mbox.$

Otherwise, continue with

:$P = - - c,$

:$Q = - + - ,$

:$R = - \pm \sqrt,$

(either sign of the square root will do)

: U = \sqrt

(there are 3 complex roots, any one of them will do)

: y = - a + \beginU=0 &\to -\sqrt \U\ne 0, &\to U - ,\end \quad\quad\quad

:$W=\sqrt$

:$x = - + .$

::The two ±_s must have the same sign, the ±_''t'' is independent. To get all roots, compute ''x'' for (±_''s'',±_''t'') = (+,+); (+,−); (−,+); (−,−). This formula handles repeated roots without problem.

Ferrari was the first to discover one of these labyrinthine solutions. 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 () 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 () has the properties that
:$C\left(\right) = < 0$ and
$\lim_ C(v) = \infty,$
where ''a'' and ''b'' are given by ().

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

Using this root the term $\sqrt$ in () is always real,
which ensures that the two quadratic equations () have real coefficients.

Obtaining alternative solutions the hard way

It could happen that one only obtained one solution through the formulae above, because not all four sign patterns are tried for four solutions, and the solution obtained is complex. It may also be the case that one is only looking for a real solution. Let ''x''₁ 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 ''x''₂ which is the complex conjugate of ''x''₁. If the other two roots are denoted as ''x''₃ and ''x''₄ 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:

and

Since
:$x_2 = x_1^\star$
then
:
\begin
(x-x_1)(x-x_2)&=x^2-(x_1+x_1^\star)x+x_1x_1^\star \\
&=x^2-2\operatorname(x_1)x+ operatorname(x_1)2+ operatorname(x_1)2.
\end

Let
:$a = - 2\operatorname(x_1),$
: b = \left \operatorname( x_1) \right + \left \operatorname(x_1) \right
so that equation () becomes

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

Multiplying equations () and () produces

Comparing equation () to the original quartic equation, it can be seen that
:$a + w = ,$
:$b + w a + v = ,$
:$w b + v a = ,$
and
:$v b = .$
Therefore
:$w = - a = + 2 \operatorname(x_1),$
:$v = = \frac.$
Equation () can be solved for ''x'' yielding
:$x_3 = ,$
:$x_4 = .$
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 substitution that is hard to memorize. Here is an approach that makes it easy to understand. The job is done if we can factor the quartic equation into a product of two quadratics. Let

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

By equating coefficients, this results in the following set of simultaneous equations:
:$\begin b & = p + r \\ c & = q + s + pr \\ d & = ps + qr \\ e & = qs \end$

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:
:$\begin c + p^2 & = s + q \\ d/p & = s - q \\ e & = sq \end$

It's now easy to eliminate both $s$ and $q$ by doing the following:
:$\begin \left(c + p^2\right)^2 - (d/p)^2 & = (s + q)^2 - (s - q)^2 \\ & = 4sq \\ & = 4e \end$

If we set $P = p^2$, then this equation turns into the cubic equation:
:$P^3 + 2cP^2 + \left(c^2 - 4e\right)P - d^2 = 0$
which is solved elsewhere. Once you have $p$, then:
:$\begin r & = -p \\ 2s & = c + p^2 + d/p \\ 2q & = c + p^2 - d/p \end$

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.

Möbius transformation method

A suitably chosen Möbius transformation can transform a quartic equation into a quadratic equation in the new variable squared. This is a known method.

Finding such a Möbius transformation involves solving a cubic equation and so simplifies the problem. For example, start with the depressed quartic equation with unity leading coefficient and with neither $a_1$ nor $a_0$ equal to zero:

$x^4+a_2x^2+a_1x+a_0=0$

and do the Möbius transformation:

$x=\frac$

Set the first and third order coefficients of the resulting quartic equation in $y$ to zero. After some algebra, one finds $A+B$ is to be obtained from the cubic equation

$a_1(A+B)^3+(4a_0-2a_1a_2-^2)(A+B)^2-2a_1a_2(A+B)-^2=0$

and, regarding $A+B$ as known, $A$ is to be obtained from the quadratic equation

$2(A+B)A^2-2(A+B)^2A-a_2(A+B)-a_1=0$
Solving the resulting quadratic equation for $y^2$ gives two values for $y^2$ and each square root of $y^2$ has two values, giving a total of four solutions, as expected.

The cubic equation in $\textbf A + \textbf B$ given earlier is the same as $P^2-Q(A+B)^2=0$, where
$P \equiv \frac=2\,A\,B\,(A+B)+a_2(A+B)+a_1$
$Q \equiv \frac=4A^2B^2-a_1(A+B)-4a_0=0$
Here ''b''_i are the coefficients of the quartic polynomial in y. This shows how this equation was obtained.

Galois theory and factorization

The symmetric group ''S''₄ on four elements has the Klein four-group as a normal 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.
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
:$\begin s_0 &= \tfrac12(r_0 + r_1 + r_2 + r_3), \\ s_1 &= \tfrac12(r_0 - r_1 + r_2 - r_3), \\ s_2 &= \tfrac12(r_0 + r_1 - r_2 - r_3), \\ s_3 &= \tfrac12(r_0 - r_1 - r_2 + r_3), \end$
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 ''s''₀ = −''b''/2, we really only need the values for s₁, s₂ and s₃. These we may find by expanding the polynomial
:$\left(z^2 - s_1^2\right)\left(z^2-s_2^2\right)\left(z^2-s_3^2\right)\qquad (2)$
which if we make the simplifying assumption that ''b'' = 0, is equal to
:$z^6 + 2cz^4 + \left(c^2-4e\right) z^2 - d^2 \qquad(3)$
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

: $F_1 = x^2+wx+\frac 1 2 w^2+\frac 1 2 c - \frac 1 2\cdot \frac -\frac 1 2 \cdot\frac - \frac + 2\frac$
: $F_2 = x^2-wx + \frac 1 2 w^2 + \frac 1 2 c + \frac 1 2\cdot \frac + \frac - 2\frac + \frac 1 2\cdot \frac$

then

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

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

Approximate methods

The methods described above are, in principle, exact root-finding methods. It is also possible to use successive approximation methods which iteratively converge towards the roots, such as the Durand–Kerner method. Iterative methods are the only ones available for quintic and higher-order equations, beyond trivial or special cases.

See also

*Linear equation
*Quadratic equation
*Cubic equation
*Quintic equation
*Polynomial
*Newton's method
* Principal equation form

References

Ferrari's achievement
*

Notes

External links

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