Several mathematical methods are developed for the solution of the cubic equations. The method of substitution or replacement of a cube of an auxiliary variable and also a number of iterative methods, in particular, Newton's method is often used. But the classical solution of the cubic equation is expressed in application of formulas of Viete and Cardano. Viete-Cardano's method is based on use of a formula of a cube of the sum of coefficients and is applicable for any kind of the cubic equation. For search of roots of the equation its record needs to be presented in the form: x ²+b*x+c=0 where an is not zero number.
Instruction
1. Write down the initial cubic equation in a look: x ²+b*x+c=0. For this purpose divide all coefficients of the equation into the first coefficient at a multiplier x³, so that it became equal to unit.
2. Proceeding from an algorithm of a method of Viete-Cardano, calculate R and Q values on the corresponding formulas: Q = (a²-3b)/9, R= (2a³-9ab+27c)/54. And coefficients of a, b and with are coefficients of the given equation.
3. Compare the received R and Q values. If right? expression Q³> R², therefore, are present at the initial equation 3 valid roots. Calculate them on Viete's formulas.
4. At values Q³ <= R², is in the decision one valid root h1 and two complex interfaced roots. For their definition it is necessary to find intermediate values A and B. Calculate them on formulas Cardano.
5. Find the first valid root on a formula x1= (B + A) - a/3. At various values A and B define interfaced roots of the complex cubic equation on the corresponding formulas.
6. If values A and B turned out equal, then the interfaced roots degenerate in the second valid root of the initial equation. It is that case when valid a root it turns out two. Calculate the second valid root on formula x2=-A-a/3.