A general quartic equation (also called a biquadratic equation) is a fourth-order polynomial equation of the form

Ferrari was the first to develop an algebraic technique for solving the general quartic, which was stolen and published in Cardano's Ars Magna in 1545 (Boyer and Merzbach 1991, p. 283). Mathematica[u] can solve quartic equations exactly using the built-in command [u]Solve[a4 x^4 + a3 x^3 + a2 x^2 + a1 x + a0 == 0, x]. The solution can also be expressed in terms of Mathematica[u] algebraic root objects by first issuing [u]SetOptions[Roots, Quartics->False].
The roots of this equation satisfy Vièta's formulas:




where the denominators on the right side are all . Writing the quartic in the standard form

the properties of the symmetric polynomials appearing in Vièta's theorem then give




Eliminating p, q, and r, respectively, gives the relations



as well as their cyclic permutations.
Ferrari was the first to develop an algebraic technique for solving the general quartic. He applied his technique (which was stolen and published by Cardano ) to the equation

(Smith 1994, p. 207).
The term can be eliminated from the general quartic (1) by making a substitution of the form

so


Letting so

then gives the standard form

where



The quartic can be solved by writing it in a general form that would allow it to be algebraically factorable and then finding the condition to put it in this form. The equation that must be solved to make it factorable is called the resolvent cubic. To do this, note that the quartic will be factorable if it can be written as the difference of two squared terms,

It turns out that a factorization of this form can be obtained by adding and subtracting (where u is for now an arbitrary quantity, but which will be specified shortly) to equation (6) to obtain

This equation can be rewritten

(Birkhoff and Mac Lane 1965). Note that the first term is immediately a perfect square with

and the second term will be a perfect square if u is chosen to that the square can be completed in

This means we want

which requires that

or

This is the resolvent cubic.

Since an analytic solution to the cubic is known, we can immediately solve algebraically for one of the three solution of equation (29), say , and plugging equation (29) into equation (26) then gives

with

Q therefore is linear in x and P is quadratic in x, so each term and is quadratic and can be solved using the quadratic formula, thus giving all four solutions to the original quartic.
Explicitly, plugging p, q, and r back into (29) gives


This can be simplified by making the substitution

which gives the resolvent cubic equation

Let be a real root of (34), then the four roots of the original quartic are given by the roots of the equation

which are




where



(Abramowitz and Stegun 1972, p. 17; Beyer 1987, p. 12).

Another approach to solving the quartic (6) defines



where the second forms follow from

and defining


This equation can be written in terms of the original coefficients p, q, and r as

The roots of this cubic equation then give ,,and , and the equations (43) to (45) can be solved for the four roots of the original quartic (Faucette 1996).