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
where a ≠ 0.
The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value).
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.^{[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 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.^{[2]}
Consider a quartic equation expressed in the form :
There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is nonzero. 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.
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,
Call our quartic polynomial Q(x). Since 1 raised to any power is 1,
Thus if Q(1) = 0 and so x = 1 is a root of Q(x). It can similarly be shown that if 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.
If and then is a root of the equation. The full quartic can then be factorized this way:
Alternatively, if and then x = 0 and x = −k become two known roots. Q(x) divided by x(x + k) is a quadratic polynomial.
A quartic equation where a_{3} and a_{1} are equal to 0 takes the form
and thus is a biquadratic equation, which is easy to solve: let , so our equation turns to
which is a simple quadratic equation, whose solutions are easily found using the quadratic formula:
When we've solved it (i.e. found these two z values), we can extract x from them
If either of the z solutions were negative or complex numbers, then some of the x solutions are complex numbers.
Steps:
This leads to:
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:^{[3]}
So, if the three nonmonic coefficients of the depressed quartic equation in terms of the five coefficients of the general quartic equation are given as follows: , and ; then, the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are exposed below.
To begin, the quartic must first be converted to a depressed quartic.
Let

(1') 
be the general quartic equation which it is desired to solve. Divide both sides by A,
The first step, if B is not already zero, should be to eliminate the x^{3} term. To do this, change variables from x to u, such that
Then
Expanding the powers of the binomials produces
Collecting the same powers of u yields
Now rename the coefficients of u. Let
The resulting equation is

(1) 
which is a depressed quartic equation.
If 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 The equation must be solved as a biquadratic.
In either case, once the depressed quartic is solved for u, substituting those values into
produces the values for x that solve the original quartic.
After converting to a depressed quartic equation
and excluding the special case b = 0, which is solved as a biquadratic, we assume from here on that b ≠ 0 .
We will separate the terms left and right as
and add in terms to both sides which make them both into perfect squares.
Let y be any solution of this cubic equation:
Then (since b ≠ 0)
so we may divide by it, giving
Then
Subtracting, we get the difference of two squares which is the product of the sum and difference of their roots
which can be solved by applying the quadratic formula to each of the two factors. So the possible values of u are:
Using another y from among the three roots of the cubic simply causes these same four values of u to appear in a different order. The solutions of the cubic are:
using any one of the three possible cube roots. A wise strategy is to choose the sign of the squareroot that makes the absolute value of w as large as possible.
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
to equation (1), yielding

(2) 
The effect has been to fold up the u^{4} term into a perfect square: (u^{2} + α)^{2}. The second term, αu^{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 (2), and a corresponding 2y into the coefficient of u^{2} in the right side. To accomplish these insertions, the following valid formulas will be added to equation (2),
and
These two formulas, added together, produce
which added to equation (2) produces
This is equivalent to

(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:
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:
Therefore to make the right side of equation (3) into a perfect square, the following equation must be solved:
Multiply the binomial with the polynomial,
Divide both sides by −4, and move the −β^{2}/4 to the right,
Divide both sides by 2,

(4) 
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.
With the value for y so selected, it is now known that the right side of equation (3) is a perfect square of the form
so that it can be folded:
Therefore equation (3) becomes

(5) 
Equation (5) 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:

(5') 
Collecting like powers of u produces

(6) 
Equation (6) is a quadratic equation for u. Its solution is
Simplifying, one gets
This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are

(6') 
Given the quartic equation
its solution can be found by means of the following calculations:
If then
Otherwise, continue with
(either sign of the square root will do)
(there are 3 complex roots, any one of them will do)
Ferrari was the first to discover one of these labyrinthine solutions^{[citation needed]}. The equation which he solved was
which was already in depressed form. It has a pair of solutions which can be found with the set of formulas shown above.
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
where P and Q are given by (5) has the properties that
where α and β are given by (1).
This means that (5) has a real root greater than , and therefore that (4) has a real root greater than .
Using this root the term in (8) is always real, which ensures that the two quadratic equations (8) have real coefficients.^{[5]}
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
but this quartic equation is equivalent to the product of two quadratic equations:

(9) 
and

(10) 
Since
then
Let
so that equation (9) becomes

(11) 
Also let there be (unknown) variables w and v such that equation (10) becomes

(12) 
Multiplying equations (11) and (12) produces

(13) 
Comparing equation (13) to the original quartic equation, it can be seen that
and
Therefore
Equation (12) can be solved for x yielding
One of these two solutions should be the desired real solution.
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 factor the quartic equation into a product of two quadratics. Let
By equating coefficients, this results in the following set of simultaneous equations:
This is harder to solve than it looks, but if we start again with a depressed quartic where , which can be obtained by substituting for , then , and:
It's now easy to eliminate both and by doing the following:
If we set , then this equation turns into the cubic equation:
which is solved elsewhere. Once you have , then:
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 for the square root of merely exchanges the two quadratics with one another.
The symmetric group S_{4} on four elements has the Klein fourgroup 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
If we now set
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 s_{1}, s_{2} and s_{3}. These we may find by expanding the polynomial
which if we make the simplifying assumption that b = 0, is equal to
This polynomial is of degree six, but only of degree three in z^{2}, 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
then
We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.
The methods described above are, in principle, exact methods which find the roots once and for all. It is also possible to use methods which give successive approximations which hopefully improve with each iteration. Once such method is the Durand–Kerner method. Such methods may be the only ones available, other than special cases, when trying to solve quintic and higher equations.