Search results
Results from the WOW.Com Content Network
This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in § Depressed cubic. The formula can be proved as follows: Starting from the equation t 3 + pt + q = 0, let us set t = u cos θ.
Cubic equations, which are polynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a rational number).
The roots , of the quadratic polynomial () = + + satisfy + =, =. The first of these equations can be used to find the minimum (or maximum) of P ; see Quadratic equation § Vieta's formulas .
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
This equation may have up to three roots. The maximal root of the cubic equation generally corresponds to a vapor state, while the minimal root is for a liquid state. This should be kept in mind when using cubic equations in calculations, e.g., of vapor-liquid equilibrium.
This is a cubic equation in y. Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.
Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced to the computation of square and cube roots. Cardano's formula for solution in ...
In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation. The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements.