# How to Solve/NSolve/Reduce this rationalized equation

In this problem I don't know why the system is not solvable. I have just one unknown K0.

   sol[K0_]:=SetPrecision[-(2397273135785684105116286138088240239609486751943174/
6428810371332877606946043738694402071216515377775) - (
7849 Sqrt[-41616 + (
149163411674835586124096205921334157174479655112799 Sqrt[K0])/
35362087636687395438079446898535340978108366770607])/1400 + (
7849 I Sqrt[
41616 + (149163411674835586124096205921334157174479655112799 Sqrt[
K0])/35362087636687395438079446898535340978108366770607])/1400 + (
1752165303140718220448962503 Sqrt[-41616 + (
149163411674835586124096205921334157174479655112799 Sqrt[K0])/
35362087636687395438079446898535340978108366770607])/(
1605413612500000000 K0) - (
1752165303140718220448962503 I Sqrt[
41616 + (149163411674835586124096205921334157174479655112799 Sqrt[
K0])/35362087636687395438079446898535340978108366770607])/(
1605413612500000000 K0) + (
15630469741744099872889401708736470294234641513962455 Sqrt[-41616 + (
149163411674835586124096205921334157174479655112799 Sqrt[K0])/
35362087636687395438079446898535340978108366770607])/(
282585461012870216280087249803974135070469452098 Sqrt[K0]) + (
15630469741744099872889401708736470294234641513962455 I Sqrt[
41616 + (149163411674835586124096205921334157174479655112799 Sqrt[
K0])/35362087636687395438079446898535340978108366770607])/(
282585461012870216280087249803974135070469452098 Sqrt[K0]) + (
5994979010910966184793439691673144599099894289304 Sqrt[
K0])/158612235779223913220534429863991677577574124084351,50];

NSolve[sol[K0] == 0, K0, WorkingPrecision -> 50]

-
It needs around 1000 digits for NSolve to get a viable result. – Daniel Lichtblau Mar 5 '14 at 15:28

You can simplify the equation by substituting

(149163411674835586124096205921334157174479655112799 Sqrt[K0])/
35362087636687395438079446898535340978108366770607 -> x


In terms of x your equation is :

    eqx = -(2397273135785684105116286138088240239609486751943174/
6428810371332877606946043738694402071216515377775) - (
7849 Sqrt[-41616 + x])/1400 + (
921717274625804815841653990450552941488955365821976159172044897066646\
5442448592066192998080942936 \
x)/1028658357487063605533427112075979998202376465461546298411614076580\
745421176464853188892913108509063 + (
7849 I Sqrt[
41616 + x])/1400 + \
(144389604871311196692588725680621072006459662286351873532593008362559\
0814659552848877038050625016396017600390479939064435177989 \
(Sqrt[-41616 + x] -
I Sqrt[41616 +
x]))/(74353080980296519773376754906763922482554647477896877573\
358888494401499022131875397519569500457054127143037500000000 x^2) + \
(259054910306537359525519986234095985067817120649321891997543630210538\
002627405673355517239942941773505 (Sqrt[-41616 + x] +
I Sqrt[41616 +
x]))/(11103124263545362076398381815222238383393706369594446557\
95531450212636408691267807637700548987054 x) ;


This can, apparently, be solved exactly, though through Root objects :

solx = Solve[eqx == 0, x] ;


However a check gives :

eqx /. N[solx, 50]
(* not sure about the first solution *)

-
thank you b.gatessucks – Pipe Mar 5 '14 at 16:50
You're welcome. Do not rush accepting as this might discourage others from providing a better (and full) solution. – b.gatessucks Mar 5 '14 at 19:18