The equations of the third degree are called still the cubic equations. It the equations in which the senior degree at variable x is the cube (3).
Instruction
1. The cubic equation in a general view looks so: ax³ + bx² + cx + d = 0, to an is not equal to 0; a, b, c, d are real numbers. By a universal method of the solution of the equation of the third degree is method Cardano.
2. For a start we lead the equation to a type of y³ + py + q = 0. For this purpose we make replacement of variable x by y - b/3a. You watch substitution of replacement in the drawing. There are two formulas of abridged multiplication used for removal of brackets: (a-b)³ = a³ - 3a²b + 3ab² - b³ and (a-b)² = a² - 2ab + b². Then we bring similar composed and we group in variable y degrees.
3. Now, to receive at y³ single coefficient, we divide all equation into a. Then we will receive the following formulas for coefficients of p and q in y equation³ + py + q = 0.
4. Then we calculate special sizes: Q, α, β which will allow to calculate equation roots with y.
5. Then three roots of the equation of y³ + py + = 0 are calculated by q on formulas in the drawing.
6. If Q> 0, then y equation³ + py + q = 0 has only one material root y1 = α + β (and two complex, find them on the corresponding formulas if it is necessary). If Q = 0, then all roots of a veshchestvenna and at least two of them coincide, at the same time α = β and roots are equal: y1 = 2α, y2 = y3 =-α.Если Q <0, veshchestvenna roots, but is necessary ability to take a root from a negative number. After finding of y1, y2 and y3 substitute them in replacement x = to y - b/3a and find roots of the initial equation.