by 'Literka'. |

Donate $2, please - it may help Literka.
Press PayPal button, please |

Cardano found a formula for a root of a cubic equation of the form

By a substitution

and dividing both sides by a3 we receive a simpler form of this equation:

for some values of u and h. Of course it will be done if we find values u and h.

A very simple computation shows that

There is no need to show how to solve it. The solution is

where w is a root of third of the number 1, defined by:

Proceeding the same way as before, which is evident even without a computation, we receive formulas for 2 other roots:

from the first assumption and | ||

from the second one. |

There is a page of Literka Cardano's formula for Cubic Equations about using of Cardano's formula.

There is a page of Literka Cubic Equations - Another Approach, where another way of finding formulas is presented.

There is a similar page of Literka about formulas for roots of quartic equations Quartic Equations. Cardano's Formulas.

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.

Software that you really need.

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 .

Weight centers of simple geometrical figures.

Rudin's Theorem of Complex Analysis.

An elementary problem can be unsolvable.

Construction of a regular pentagon.

Construction of a regular heptadecagon.

Construction of a regular polygon with 257 sides.

Weierstrass Approximation Theorem. Bernstein's Polynomials.

Exact values of trigonometric functions of angles (n*pi)/7.

Exact values of trigonometric functions of angles (n*pi)/11.

Exact values of trigonometric functions of angles (n*pi)/13.

Exact values of cos(k*pi)/17.

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, Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.

Positive random walks.

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'.