# Reduce return a strange result for solving a equation with radical

Bug introduced in 9.0 or earlier and fixed in 11.3

Reduce[1/Sqrt[x] == x + 1/(Sqrt + Sqrt), x]


the output is unreadable and include the symbol ReduceCADAlgVar.

• Same in all versions between 9-10.2. But this is not a question. If you think you found a bug, please contact Wolfram Support. Posting here won't ensure that they will even see it. – Szabolcs Aug 19 '15 at 13:21
• Confirm unexpected output on Win7-32/MMA 10.1. Given the community confirmations I've added the bugs tag. – Sjoerd C. de Vries Aug 19 '15 at 13:28
• Possibly related? Why doesn't Roots work on a certain quartic polynomial equation? – MarcoB Aug 19 '15 at 16:08
• The same bug even in Mathematica 7.0.1 on Win7x64. – innaiz Mar 6 '17 at 11:46

Restrict the domain to Reals

$Version  "10.2.0 for Mac OS X x86 (64-bit) (July 7, 2015)" eqn = 1/Sqrt[x] == x + 1/(Sqrt + Sqrt); sol = Reduce[eqn, x, Reals] // ToRules  {x -> Root[1 - 10*#1^2 - 4*#1^3 + #1^4 + 20*#1^5 + 6*#1^6 - 10*#1^8 - 4*#1^9 + #1^12 & , 4]^2} eqn /. sol // FullSimplify  True sol // N  {x -> 0.800116} eqn /. (sol // N)  True EDIT: Update for v11.3 $Version

(* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *)

eqn = 1/Sqrt[x] == x + 1/(Sqrt + Sqrt);

sol1 = Reduce[1/Sqrt[x] == x + 1/(Sqrt + Sqrt), x] // ToRules

(* {x -> Root[{-3 + #1^2 &, -2 + #2^2 &, -1 + 5 #3 - 2 #1 #2 #3 + 2 #1 #3^2 -
2 #2 #3^2 + #3^3 &}, {2, 2, 1}]} *)


Verifying,

eqn /. sol1 // RootReduce

(* True *)


Restricting the domain to Reals gives the same result as with the earlier version

sol2 = Reduce[1/Sqrt[x] == x + 1/(Sqrt + Sqrt), x, Reals] // ToRules

(* {x -> Root[
1 - 10 #1^2 - 4 #1^3 + #1^4 + 20 #1^5 + 6 #1^6 - 10 #1^8 -
4 #1^9 + #1^12 &, 4]^2} *)


The different Root objects are equivalent

(x /. sol1) == (x /. sol2) // RootReduce

(* True *)


The following works in MMA 11.3.

ToRadicals[Reduce[1/Sqrt[x] == x + 1/(Sqrt + Sqrt), x]]
`

x == 2/3 (Sqrt - Sqrt) + 1/( 3 (2/(27 - 22 Sqrt + 18 Sqrt + 3 Sqrt[3 (27 - 44 Sqrt + 36 Sqrt)]))^(1/3)) + 1/3 (5 - 2 Sqrt) (2/( 27 - 22 Sqrt + 18 Sqrt + 3 Sqrt[3 (27 - 44 Sqrt + 36 Sqrt)]))^(1/3)