Cubic Equations

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Canonical form:

ax3+bx2+cx+d=0

Reduced form:

After substitution below:

 x=y-b3a

Reduced form of Cubic Equations:

x3+3py+2q=0

Discriminant: D=q2+p3

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=-q+D3 v=-q-D3y1=u+v y2=ε1u+ε2vy3=ε2u+ε1vε1,2=-12±32i

Vieta's formulas

x1+x2+x3=-bax1·x2·x3=-da1x1+1x2+1x3=-cd

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