cubics and quartics
- Subject: cubics and quartics
- From: Jim FitzSimons
- Date: Wed, 23 Aug 2006 00:22:22 -0700
DERIVE does cubics and quartics.
SOLVE(x^3+d*x+e=0,x)
x=2*SQRT(3)*SQRT(-d)*COS(ACOS(-3*SQRT(3)*e/(2*(-d)^(3/2)))/3)/3 OR
x=-2*SQRT(3)*SQRT(-d)*SIN(ASIN(3*SQRT(3)*e/(2*(-d)^(3/2)))/3+pi/3)/3 OR
x=2*SQRT(3)*SQRT(-d)*SIN(ASIN(3*SQRT(3)*e/(2*(-d)^(3/2)))/3)/3
SOLVE(x^4+c*x^2+d*x+e=0,x)
x=-SQRT(6)*SQRT(-SQRT(c^2+12*e)*COS(ACOS(-(2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e
)~
^(3/2)))/3)-c)/6-3*d/(4*SQRT(SQRT(c^2+12*e)*SIN(ASIN((2*c^3-72*c*e+27*d^2)/(
2~
*(c^2+12*e)^(3/2)))/3+pi/3)-c)*SQRT(-SQRT(c^2+12*e)*COS(ACOS(-(2*c^3-72*c*e+
2~
7*d^2)/(2*(c^2+12*e)^(3/2)))/3)-c))-SQRT(6)*SQRT(SQRT(c^2+12*e)*SIN(ASIN((2*
c~
^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3+pi/3)-c)/6 OR
x=-SQRT(6)*SQRT(-SQRT~
(c^2+12*e)*COS(ACOS(-(2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3)-c)/6+3*d
/~
(4*SQRT(SQRT(c^2+12*e)*SIN(ASIN((2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/
3~
+pi/3)-c)*SQRT(-SQRT(c^2+12*e)*COS(ACOS(-(2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)
^~
(3/2)))/3)-c))+SQRT(6)*SQRT(SQRT(c^2+12*e)*SIN(ASIN((2*c^3-72*c*e+27*d^2)/(2
*~
(c^2+12*e)^(3/2)))/3+pi/3)-c)/6 OR
x=SQRT(6)*SQRT(-SQRT(c^2+12*e)*COS(ACOS(-(~
2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3)-c)/6-3*d/(4*SQRT(SQRT(c^2+12*e
)~
*SIN(ASIN((2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3+pi/3)-c)*SQRT(-SQRT(
c~
^2+12*e)*COS(ACOS(-(2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3)-c))+SQRT(6
)~
*SQRT(SQRT(c^2+12*e)*SIN(ASIN((2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3+
p~
i/3)-c)/6 OR
x=SQRT(6)*SQRT(-SQRT(c^2+12*e)*COS(ACOS(-(2*c^3-72*c*e+27*d^2)/(~
2*(c^2+12*e)^(3/2)))/3)-c)/6+3*d/(4*SQRT(SQRT(c^2+12*e)*SIN(ASIN((2*c^3-72*c
*~
e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3+pi/3)-c)*SQRT(-SQRT(c^2+12*e)*COS(ACOS(-(2
*~
c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3)-c))-SQRT(6)*SQRT(SQRT(c^2+12*e)*
S~
IN(ASIN((2*c^3-72*c*e+27*d^2)/(2*(c^2+12*e)^(3/2)))/3+pi/3)-c)/6
Maple does cubics but not quartics.
solve(x^3+d*x+e=0,x);
1/6*(-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3)-2*d/(-108*e+12*(12*d^3+81*e^2)^(
1/2))^(1/3),
-1/12*(-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3)+d/(-108*e+12*(12*d^3+81*e^2)^(
1/2))^(1/3)+1/2*I*3^(1/2)*(1/6*(-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3)+2*d/(
-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3)),
-1/12*(-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3)+d/(-108*e+12*(12*d^3+81*e^2)^(
1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3)+2*d/(
-108*e+12*(12*d^3+81*e^2)^(1/2))^(1/3))
solve(x^4+c*x^2+d*x+e=0,x);
RootOf(_Z^4+c*_Z^2+d*_Z+e)
Regards, Jim FitzSimons