# Cubic Equations

Keywords:

Canonical form:

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

Reduced form:

After substitution below:

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

Vieta's formulas

${x}_{1}+{x}_{2}+{x}_{3}=-\frac{b}{a}\phantom{\rule{0ex}{0ex}}{x}_{1}·{x}_{2}·{x}_{3}=-\frac{d}{a}\phantom{\rule{0ex}{0ex}}\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}+\frac{1}{{x}_{3}}=-\frac{c}{d}$