Cubic equation

ax ³ + bx ² + cx + d = 0. - Cubic equation.

Solution of the cubic equation:

1. The decision of Cardano .

Roots of an incomplete cubic equation
y ³ + py + q = 0
expressed by the formulas:


Where
,

moreover, A and B are the values ​​of the corresponding roots, for example AB = -p / 3.

The number of real roots of the cubic equation depends on the sign of the discriminant D:

D> 0 is one real root and two conjugate complex roots.

D <0 - three real roots.

D = 0 - one single real root and two double, or, if p = q = 0, then one triple real root.

2. Trigonometric solution

If the coefficients p and q of an incomplete cubic equation are real, then its roots can be expressed in terms of trigonometric functions:

a) Let p <0 and D <0, then



where the trigonometric functions are expressed as:



b) Let p> 0 and D ≥ 0, then



where trigonometric functions are expressed as:



c) Let p <0 and D ≥ 0, then:



where trigonometric functions are expressed as:



In all these cases, the actual values ​​of the cube roots are taken.

3. The roots of the cubic equation ax ³ + bx ² + cx + d = 0 are expressed by the formulas:



where y _k is the roots of an incomplete cubic equation with coefficients:



Vieta theorem for the roots of a complete cubic equation: