Roots of a Cubic Equation.
Cardano's Formula.


by 'Literka'.




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Cardano found a formula for a root of a cubic equation of the form
There are many pages on the Internet with a proof of Cardano's formula. Literka decided to add one more, since this proof seems for Literka to be a little more straightforward. However this is left to our viewers to judge.

By a substitution

and dividing both sides by a3 we receive a simpler form of this equation:
These are routine procedures you can find on most of pages about Cardano's formula. Now we'll do something different. We'll try to guess the value of z and our guess is that


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
which is exactly what we wanted if

There is no need to show how to solve it. The solution is
Substituting these equalities we receive the formula of Cardano
for the equation
This is one root of the above cubic equation. The other two roots can be found similar way with a little different assumption about z:

or we take


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.



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