Roots of Quartic Equations
Cardano's Formula

Let us consider, for rational numbers a,b,c,d, an equation


be the roots of this equation and let L be a field generated by the set Q of rational numbers and these roots.
The Galois group Gal L/Q of this equation is a group S4 of all permutation of four elements. We assume that these elements are roots x1, x2, x3, x4.
The discriminant of this equation is a number s defined by

The expression is invariant under acting of the group Gal L/K, hence it must be rational. In fact, we can express s in terms of coefficients of the equation. Number s (discriminant) is equal

Dropping powers in a formula defining s, we can easily verify that

Let us define


We can verify that

Applying formulas for symmetric functions and formulas of Vieta, we see that

All last equalities, they are formulas of Vieta of the equation

It follows that A1, A2, A3 are the roots of this equation. Formulas of Cardano for roots of cubic equation give us

where w is a root of 1 of third degree defined by

and u is a number equal

From the definition of the numbers A1, A2, A3 and a simple computation it follows that

Taking square roots, adding one equality from formulas of Vieta, we receive a system of four linear equations with four unknowns

Solving this system, we receive final formulas

Click here to see a similar page of Literka for a solution of cubic equation.
If you are interested in mathematical software about cubic equations click new module "Cubic Functions" of a free program "Ruler and Compass" or download this program.
There is something like an eBook about quartic and cubic equations. See "Mathematical Reader" to learn more.

See the list and descriptions of mathematical pages from Mathematical Countryside.

See other pages of 'Literka' from Mathematical Countryside:
Monotonic subsequences.
A remarkable monotonic property of the gamma function .
An elementary problem can be unsolvable.
Weierstrass Approximation Theorem. Bernstein's Polynomials.
Construction of a regular pentagon.
Roots of cubic equation. Cardano's formula.
Rudin's Theorem of Complex Analysis.
Exact values of trigonometric functions of angles (n*pi)/17.
Equalities for values of trigonometric functions of angles (n*pi)/17.
Factorization of a polynomial, which defines values of sine function (angles n*pi/17).
Polynomials with roots cos(2*k*pi/n).
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
Values of trigonometric functions of angles (n*pi)/257. Part I.
Values of trigonometric functions of angles (n*pi)/257. Part II, Part III, Part IV, Part V, Part VI, Part VII.
Values of trigonometric functions of angles (n*pi)/65537. Part I.
Values of trigonometric functions of angles (n*pi)/65537. Part II, Part III, Part IV and Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.

Construction of a regular heptadecagon.
Construction of a regular polygon with 257 sides.

Software that you really need.

Return to the list of pages of 'Literka' about polytopes.
Return to the main geometrical page of 'Literka'.
Return to the main page of 'Literka'.