CMH117@PSUVM.BITNET (Charles Hannum) (11/10/89)
I mistakenly thought that I was going to weedle my way out of looking
this up again. Here is the official solution to the quartic equation,
known as (appropriately enough) "Ferrari's (Ferraro's) solution to the
quartic." Gee, aren't these mathematicians imaginative?
We'll start with "The Universal Encyclopedia of Mathematics."
A biquadratic or quartic equation is an equation of the form:
4 3 2
x + ax + bx + cx + d = 0
By the transformation
a
x = z - -
4
this can be reduced to the standard form
4 2
x = z + pz + qz + r = 0
If we form the cubic resolvent
2 2
3 p 2 p - 4r q
y + - y + ------- y - -- = 0
2 16 64
and determine its solutions y1, y2, y3, we may obtain the four solutions
of the reduced equation from
z = +- sqrt(y1) +- sqrt(y2) +- sqrt(y3)
where the signs are chosen so that
q
sqrt(y1)*sqrt(y2)*sqrt(y3) = - -
8
Now that I've muddled this group with mathematics ... B-)
For those of you who don't want to try to solve the previous equation,
the next selection is from a book whose title I can't remember ...
FERRARI (or FERRARO), Ludovico (1522-
1565). Ferrari's solution to the quartic. The
solution of the quartic equation
4 3 2
x + px + qx + rx + s = 0
by showing that the roots of this equation
are also the roots of the two equations
2 1
x + - px + k = +- (ax + b),
2
where
1 2 (1/2)
a = (2k + - p - q)
4
b = (kp - r) / (2a)
and k is obtained from the following cubic
equation (the RESOLVENT CUBIC):
3 1 2 1 1 2 2
k - - qk + - (pr - 4s) k + - (4qs - p s - r ) = 0
2 4 8
Notes: The RESOLVENT CUBIC above can be solved with Cardan's formula.
The symbol "+-" means "plus or minus."
I've gone through a *LOT* of trouble to make sure the mathematics
didn't get mumbled as I typed them in. As of the time I press the
SEND key, this information is correct. I don't guarantee what
will happen to it after it is sent.
When implemented correctly, this yields a fast, accurate result.
--
- Charles Martin Hannum II "Klein bottle for sale... Inquire within."
(and PROUD OF IT!!!) "To life immortal!"
c9h@psuecl.psu.edu "No noozzzz izzz netzzznoozzzzz..."
cmh117@psuvm.psu.edu "Memories, all alone in the moonlight ..."