Interpreting the solutions

• To: mathgroup at smc.vnet.net
• Subject: [mg75112] Interpreting the solutions
• From: "Apostolos E. A. S. Evangelopoulos" <a.e.a.evangelopoulos at sms.ed.ac.uk>
• Date: Wed, 18 Apr 2007 04:50:04 -0400 (EDT)

```Hello,

(8*Pi*R^3*S)/(3*h^2) + (2*h*Pi*(S + 3*Î³))/3 +
((R^2*(-h + 2*R)^2*(-h/(2*R^2) - (2*R)/h^2))/
(Sqrt[3]*Sqrt[R^2*(-h^2/(4*R^2) + (2*R)/h)]) -
(4*(-h + 2*R)*Sqrt[R^2*(-h^2/(4*R^2) + (2*R)/h)])/Sqrt[3] -
(8*R*(-h + 2*R)*(-h/(2*R^2) - (2*R)/h^2)*Sqrt[R^2*(-h^2/(4*R^2) + (2*R)/h)])/
(3*Sqrt[3]) + (16*(R^2*(-h^2/(4*R^2) + (2*R)/h))^(3/2))/(9*Sqrt[3]*R) +
(16*(-h/(2*R^2) - (2*R)/h^2)*(R^2*(-h^2/(4*R^2) + (2*R)/h))^(3/2))/(9*Sqrt[3]))*Î? == 0

with respect to h and the solutions -after about 3/4 of an hour of calculations(!)- came up as

\!\({{h -> Root[\(-32768\)\ R\^17\
Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\
Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\
Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ #1\^4 +
1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 - 76608\ R\^12\ Î?\^2\ #1\^5 + \
3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + 40608\
R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ Ï?\^2\
\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\
Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ \
R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\
Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\
\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\
Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - \
13456\ R\^7\ Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\
R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\
\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\
Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ \
#1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 1]}, {h -> Root[\(-32768\)\ \
R\^17\ Î?\^2 + 81920\
R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\
\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ \
#1\^4 + 1728\ Ï?\^2\
R\^10\ S\^2\ #1\^5 - 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ \
R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ \
#1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\
S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ \
S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\
\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\
Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ \
#1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\
\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ \
S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\
Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ \
#1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 +
114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, \
2]}, {h -> Root[\(-32768\)\
R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\
Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ \
R\^11\ S\^2\ #1\^4 +
73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 - \
76608\ R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\
\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + \
5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ \
S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\
Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\
Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ \
R\^5\ S\^2\ #1\^10 + 1296\
Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - \
13456\ R\^7\ Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ \
S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - \
4494\ R\^5\ Î?\^2\ #1\^12 + 1921\
R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\
Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 3]}, {h ->
Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\
R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\
\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ \
#1\^5 - 76608\
R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ \
Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\
S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\
Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\
\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\ \
Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\
Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ \
S\ Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\
Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\
S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\
\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\
\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, \
4]}, {h -> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\
Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\
R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\
R\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ #1\^4 +
1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 - 76608\ R\^12\ Î?\^2\ #1\^5 + \
3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + 40608\ R\^11\
\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 \
- 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\
\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\
Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ \
Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ \
#1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 +
1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + \
108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ \
R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 +
1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ \
Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 5]}, {h -> Root[\(-32768\
\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\
Î?\^2\ #1\^2 + 6912\ Ï?\^2\
R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\
\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\
R\^10\ S\^2\ #1\^5 - 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\
Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + \
40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\
S\ Î³\ #1\^7 - 18608\ R\^10\
Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\
Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\
\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\
R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\
S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\ Ï?\^2\ \
R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\
Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\
S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\
\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\
\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, \
6]}, {h -> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\
R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\
R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\
R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 - 76608\ \
R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ \
#1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ \
Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ \
#1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\
Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ \
Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ \
#1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 +
1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + \
108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ \
R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 -
4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\
R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\
R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 7]}, {h -> Root[\(-32768\)\ R\
\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ \
R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 +
73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 \
- 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\
Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\
\^2\ R\^8\ S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\
Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\
Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\
\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\
R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\
S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\ Ï?\^2\ \
R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ \
#1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\
\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - \
564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 +

Î?\^2\ #1\^17 &, 8]}, {h -> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\
\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\
R\^12\ S\^2\ #1\^3 + 3456\
Ï?\^2\ R\^11\ S\^2\ #1\^4 +
73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 \
- 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ \
R\^9\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 \
+ 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\
\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\
Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\^6\ S\^2\ \
#1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\
\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ \
#1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + 108\ \
Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\
\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\
R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\
Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 9]}, {h \
-> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\
\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 +
73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 \
- 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\
Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 +
40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + \
5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ \
S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\
\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\
R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\
\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ \
#1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\
Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\
Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\
\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - \
564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + \
Î?\^2\ #1\^17 &, 10]}, {h ->
Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ \
R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\
R\^11\ S\^2\ #1\^4 + 73600\ R\^13\
Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 - 76608\ R\^12\
\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\ S\ Î³\ \
#1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ \
Ï?\^2\ R\^8\
S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\
\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ \
R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ \
#1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\
\ R\^5\ S\ Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\
\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\
\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\
Î?\^2\ #1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\
Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + \
Î?\^2\ #1\^17 &, 11]}, {h -> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\
\ #1 - 86016\
R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\ \
Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\
S\^2\ #1\^5 - 76608\ R\^12\
Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\
\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + \
5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 -
18608\ R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\
\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\
R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ \
R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\
Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\
Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - \
13456\ R\^7\ Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ \
S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 +
8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ \
R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 +
114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, \
12]}, {h -> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ \
R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\
Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ \
Ï?\^2\ R\^10\ S\^2\ #1\^5 - 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\
Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ R\^9\
S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\
S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ \
Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - \
224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\
Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ \
#1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\
Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ \
#1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ \
Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ \
#1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\
Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &,
13]}, {h -> Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\
Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ \
#1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ \
Ï?\^2\ R\^10\
S\^2\ #1\^5 - 76608\ R\^12\
Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 + 10368\ Ï?\^2\ \
R\^9\ S\ Î³\ #1\^6 +
40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 +
5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\ Î?\^2\ #1\^7 + 864\
\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\
Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\
R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ \
R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\
R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\
Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + 108\
Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ \
Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ \
#1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\
\ #1\^15 - 14\
R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 14]}, {h -> Root[\(-32768\)\ \
R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 -
86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + \
3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\
R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 -
76608\
R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 +
10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 +
40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\
S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\
Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\
S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 +
432\ Ï?\^2\ R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\
Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ \
#1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\
Î³\ #1\^10 + 1944\ Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ \
#1\^10 + 108\ Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ \
Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ \
#1\^12 + 1921\ R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\
\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 15]}, {h -> Root[\(-32768\)\ \
R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ #1\^2 + 6912\ Ï?\^2\
\ R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + 73600\
R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\
S\^2\ #1\^5 - 76608\ R\^12\ Î?\^2\ #1\^5 + 3456\
Ï?\^2\ R\^9\
S\^2\ #1\^6 +
10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 + 40608\ R\^11\ Î?\^2\ #1\^6 \
+ 1728\ Ï?\^2\ R\^8\ S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ \
R\^10\ Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\
Ï?\^2\ R\^7\ S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\
R\^6\ S\^2\ #1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ \
R\^6\ Î³\^2\ #1\^9 + 14896\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\
R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ #1\^10 + 1944\
Ï?\^2\ R\^5\ Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + 108\
Ï?\^2\ R\^4\ S\^2\ #1\^11 + 648\ Ï?\^2\ R\^4\ S\ Î³\ #1\^11 + 972\ \
Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\ R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ \
#1\^12 + 1921\
R\^4\ Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ \
Î?\^2\ #1\^15 - 14\ R\ Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 16]}, {h -> \
Root[\(-32768\)\ R\^17\ Î?\^2 + 81920\ R\^16\ Î?\^2\ #1 - 86016\ R\^15\ Î?\^2\ \
#1\^2 + 6912\ Ï?\^2\ R\^12\ S\^2\ #1\^3 + 3456\ Ï?\^2\ R\^11\ S\^2\ #1\^4 + \
73600\ R\^13\ Î?\^2\ #1\^4 + 1728\ Ï?\^2\ R\^10\ S\^2\ #1\^5 - 76608\
R\^12\ Î?\^2\ #1\^5 + 3456\ Ï?\^2\ R\^9\ S\^2\ #1\^6 +
10368\ Ï?\^2\ R\^9\ S\ Î³\ #1\^6 +
40608\ R\^11\ Î?\^2\ #1\^6 + 1728\ Ï?\^2\ R\^8\
S\^2\ #1\^7 + 5184\ Ï?\^2\ R\^8\ S\ Î³\ #1\^7 - 18608\ R\^10\
Î?\^2\ #1\^7 + 864\ Ï?\^2\ R\^7\ S\^2\ #1\^8 + 2592\ Ï?\^2\ R\^7\
S\ Î³\ #1\^8 - 224\ R\^9\ Î?\^2\ #1\^8 + 432\ Ï?\^2\ R\^6\ S\^2\ \
#1\^9 + 2592\ Ï?\^2\ R\^6\ S\ Î³\ #1\^9 + 3888\ Ï?\^2\ R\^6\ Î³\^2\ #1\^9 + 14896\
\ R\^8\ Î?\^2\ #1\^9 + 216\ Ï?\^2\ R\^5\ S\^2\ #1\^10 + 1296\ Ï?\^2\ R\^5\ S\ Î³\ \
#1\^10 + 1944\ Ï?\^2\ R\^5\
Î³\^2\ #1\^10 - 13456\ R\^7\ Î?\^2\ #1\^10 + 108\ Ï?\^2\ R\^4\ \
S\^2\ #1\^11 + 648\ Ï?\^2\
R\^4\ S\ Î³\ #1\^11 + 972\ Ï?\^2\ R\^4\ Î³\^2\ #1\^11 + 8028\
R\^6\ Î?\^2\ #1\^11 - 4494\ R\^5\ Î?\^2\ #1\^12 + 1921\ R\^4\ \
Î?\^2\ #1\^13 - 564\ R\^3\ Î?\^2\ #1\^14 + 114\ R\^2\ Î?\^2\ #1\^15 - 14\ R\ \
Î?\^2\ #1\^16 + Î?\^2\ #1\^17 &, 17]}}\)

(this is the output copied and pasted -sorry couldn't change it). How do I interpret this form? What ARE the solutions?

What is more, if I decide to substitute real values for all the constants in the original polynomial (apart from h, with respect to which i'm solving), and, therefore, solve something like

0.84*((175*(350 - h)^2*(-350/h^2 - h/61250))/(Sqrt[3]*Sqrt[350/h - h^2/122500]) -
(700*(350 - h)*Sqrt[350/h - h^2/122500])/Sqrt[3] -
(245000*(350 - h)*(-350/h^2 - h/61250)*Sqrt[350/h - h^2/122500])/(3*Sqrt[3]) +
(490000*(350/h - h^2/122500)^(3/2))/(9*Sqrt[3]) +
(85750000*(-350/h^2 - h/61250)*(350/h - h^2/122500)^(3/2))/(9*Sqrt[3])) +
(4544750000*Pi)/(3*h^2) + (530*h*Pi)/3 ==0,

Solve::verif : Potential solution {h-->349.9999999...344`30.7840...763} (possibly discarded by verifier) should be checked by hand. May require use of limits.
h->{-137.365 - 286.999i}, {-137.365 + 286.999i}, {h->96.8126} and a few other similar complex ones.

This set of solutions seems more reasonable, even though I don't understand the ` sign in the beginning and I guess the solution which was discarded must have created some singularity that Mathematica didn't like. In principle, the above set of solutions should be recoverable upon substitution of the constants' values in the first set of solutions.. which I do not understand, however.