# 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

$${x}_{1}+{x}_{2}+{x}_{3}=-\frac{b}{a}$$

$${x}_{1}\xb7{x}_{2}\xb7{x}_{3}=-\frac{d}{a}$$

$$\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}}=-\frac{c}{d}$$

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