bump for a simplish but kind of interesting maths puzzle

I am thinking of an integer > 10, which I will call x.

Let a be the conversion of x expressed as a number in base 10 to hexadecimal.

Let b be the conversion of x expressed as a number in hexadecimal back to base 10.

x is the geometric mean of a (expressed as a string of digits in base 10) and b.

Solve for x. There should be 2 solutions (thanks, Simon!).

To clarify, if, for example, you thought x = 24:

a = the representation of 24 in base 16, which is 0x18 (or 18).

b = the representation of 24, as a number in base 16, as a number in base 10, which is 36.

We want x to equal the geometric mean of 18 and 36, both expressed in base 10. Since 24 is not the geometric mean of 18 and 36, this is not the answer.

EDIT: @Simon's last puzzle, this is definitely not a rigorous solution but strikes me as the major block here:

The conjecture is true iff every step ends with a string of two different letters. The string starts with AB, so no appending will add two of the same letter to the end of the previous block. The string appended will always end with the pattern AB, BA, AB, BA, AB, BA, etc. satisfying the first statement.

Rigor can bite me