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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
, and
, 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
, then at least one root is
:
- if , then is a simple root;
- if , then is a double root;
- if
E = 0, D = 0, C = 0, B ≠ 0 |
, then is a triple root;
- if
E = 0, D = 0, C = 0, B = 0 |
, then is a quadruple root.
If
, 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