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The cubic formula gives the roots of any cubic equation
-
The three (distinct or not) roots are given by (Cardano published a similar formula in 1545)
-
where
-
-
-
-
Equivalently, the three (distinct or not) roots of
-
with
and
, may be written as
where
Completing the cube
The three roots are obtained by completing the cube, i.e.
or, letting
,
yielding the depressed cubic equation
with
and
.
Solving the depressed cubic equation with Vieta's substitution
The depressed cubic equation
-
is solved by doing Vieta's substitution
-
yielding
-
Multiplying by
, it becomes a
sextic equation in
, which happens to be a quadratic equation in
-
which is solved with the quadratic formula to get
-
with
-
If
,
and
are the three
cube roots for each of the two solutions in
, then
-
and
Thus, the three (distinct or not) roots are given by (see how the three roots of the two solutions are used in pairs, yielding three roots)
where
, since
.
Finally, the three (distinct or not) roots of
a x 3 + b x 2 + c x + d = 0, a ≠ 0, |
may be written as
where
with
and
.
Vieta's formulas for the cubic
Vieta's formulas for the cubic
-
gives a system of three equations in three variables (which are the three roots)
By dividing the third equation into the second equation, one obtains a formula for the harmonic sum of the roots
See also
External links