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Quartic formula

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The quartic formula gives the roots of any quartic equation

The four (distinct or not) roots are given by

where

(...)

Equivalently, the four (distinct or not) roots of

with 
B =  − 
b
4a
, C =
c
6a
, D =  − 
d
4a
, and 
E =
e
a
, may be written as

where

(...)

Vieta's formulas for the quartic

Vieta's formulas for the quartic

gives a system of four equations in four variables (which are the four roots)

If 
E = 0
, then at least one root is 
0
:
  • if 
    E = 0, D   ≠   0
    , then 
    0
    is a simple root;
  • if 
    E = 0, D = 0, C   ≠   0
    , then 
    0
    is a double root;
  • if 
    E = 0, D = 0, C = 0, B   ≠   0
    , then 
    0
    is a triple root;
  • if 
    E = 0, D = 0, C = 0, B = 0
    , then 
    0
    is a quadruple root.
If 
E   ≠   0
, then dividing the fourth equation into the third equation, one obtains a formula for the harmonic sum of the roots, and dividing the fourth equation into the second equation, one obtains a formula for the harmonic sum of the products of root pairs

See also