How to simplify the $Sqrt$ [closed]

I have a expression:

13 + 6 Sqrt[6 - x] x == x


I want to simplify the Sqrt to be -169+26 x+215 x^2-36 x^3==0.But Simplify[13 + 6 Sqrt[6 - x] x == x, Assumptions -> x < 6] don't work.And I have tried ComplexityFunction:

FullSimplify[13 + 6 Sqrt[6 - x] x == x,
ComplexityFunction -> (1000 Count[#, _Sqrt, {0, Infinity}] +
LeafCount[#] &)]


I can get anything still.So how to elminate the Sqrt?

• I don't understand. How did you obtain the second expression from the first? They don't seem to have the same solutions either. Commented Feb 21, 2017 at 14:52
• @MarcoB Confused too.I just want elminate the Sqrt.I get it by ((x-13)/6x)^2+x-6==0//FullSimplify//Expand
– yode
Commented Feb 21, 2017 at 15:06
• OK but that still didn't explain the relationship between the expression with the Sqrt and the equation you have in comments. How did you get from ((x - 13) / 6 x)^2 + x - 6 == 0 to 13 + 6 Sqrt[6 - x] x == x? Commented Feb 21, 2017 at 15:08
• @MarcoB Move  x - 6 to right of the equal sign,then execute sqrt...and so on
– yode
Commented Feb 21, 2017 at 15:12
• It should be (x-13)/(6x), not (x-13)/6x. Commented Feb 21, 2017 at 15:40

Perhaps:

Solve[13 + 6 Sqrt[6 - x] x == x, x, Cubics -> False]


{{x -> Root[169 - 26 #1 - 215 #1^2 + 36 #1^3 &, 1]}}

• little wired. :)Thanks all the same.
– yode
Commented Feb 22, 2017 at 2:15

Adapting the F function from DSolve misses a solution of a differential equation, we get

Clear[rat];
(* rationalize fractional powers *)
rat[eqn_Equal] := rat[Subtract @@ eqn] == 0;
rat[fn_] := Module[{u},
With[{rads = DeleteDuplicates@Cases[fn, Power[a_, b_Rational] :> u[a, b], Infinity]},
First@GroebnerBasis[
Flatten@{fn /. Power[a_, b_Rational] :> u[a, b],
Map[
Numerator@Together[           (*gets rid of denominators in neg.powers*)
#^Denominator[Last@#] - First[#]^Numerator[Last@#]
] &,

rat[13 + 6 Sqrt[6 - x] x == x]

(* Out[79]= 169 - 26 x - 215 x^2 + 36 x^3 == 0 *)

• Little long...;)
– yode
Commented Feb 21, 2017 at 16:15
• @yode I think you mean "too general." :) Commented Feb 21, 2017 at 16:19

GroebnerBasis can do this:

First[GroebnerBasis[13 + 6 Sqrt[6 - x] x - x, x]]


169-26 x-215 x^2+36 x^3