How to check if an expression contains a symbol ONLY after a given head?

Question

I have just started programming in Mathematica and still have to get a hang of it (and rule-based programming), so I apologize if this question may seem stupid. I will explain it with a concrete example.

I have an expression which is an inequality, and I have to determine if the unknown variable appears under a square root (the expression might also not contain any square roots at all).

So, my first thought was to use FreeQ:

FreeQ[expr, Sqrt[x]]

The problem occurs when x appears at deeper levels of the expression tree under Sqrt.

For example,

expr = 3 Sqrt[2 x + 9] + 23/ 7 x^2 >= 0

so the FullForm is

GreaterEqual[Plus[..., Times[3, Power[Plus[9, Times[2, x]], Rational[1, 2]]]], 0]

So I can't use Sqrt[___x___] or x_Sqrt as it's head is Times, and not Power.

Is there anyway I can build a pattern to use with FreeQ or is there any other function that can determine if a given expression contains a symbol AFTER a given head (e.g., Power[(something with x), 1/2] as in my case).

Answer 1

Here are two ways to do it taken from comments to the question.

kglr and J.M.

With[{expr = 3 Sqrt[2 x + 9] + Sqrt[a + 2] + 23/7 x^2 >= 0}, 
  Cases[expr, Sqrt[something_] /; ! FreeQ[something, x], ∞]]

kglr

With[{expr = 3 Sqrt[2 x + 9] + Sqrt[a + 2] + 23/7 x^2 >= 0}, 
  Cases[expr, Power[something : __, Rational[1, 2]] /; Not[FreeQ[something, x]], ∞]]