Cubic Equations

Canonical form:

$$a{x}^3+b{x}^2+cx+d=0$$

Reduced form:

After substitution below:

$$x=y-\frac{b}{3a}$$

Reduced form of Cubic Equations:

$$x^3+3py+2q=0$$

Discriminant: $$D=q^2+p^3$$

Ha D < 0: three different solutions
Ha D = 0: three real solutions, one of which is double
Ha D > 0: one real and two complex radicals

Solutions of a cubic equation

$$u=\sqrt[3]{-q+\sqrt{D}}$$

$$v=\sqrt[3]{-q-\sqrt{D}}$$

$$y_1=u+v$$

$$y_2={\epsilon }_{1}u+{\epsilon }_{2}v$$

$$y_3={\epsilon }_{2}u+{\epsilon }_{1}v$$

$${\epsilon }_{1,2}=-\frac{1}{2}\pm \frac{\sqrt{3}}{2}i$$

Vieta's formulas