Roots of Quartic Equations
Cardano's Formula


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

Let

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

Then
and

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.
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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.

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