TITLE: Looking for solution of a 9th grader's problem
QUESTION [5 upvotes]: My 9th grade (USA) son was given the following problem as part of an exam (which flummoxed him and me):
Solve $3(x-2)^2+5 = 3^{x+2} +5$.
This can be reduced to something like 
$y^2 3^y = 3^3$, but any solution in elementary functions escapes me.
Is it straightforward to show that a solution using elementary functions does not exist?
Clarification:
The problem has a unique solution which can be found easily by numerical
techniques, but this is not something I would expect on a 9th grader's final exam.
Clarification:
I am trying to determine if a solution exists in terms of elementary functions. There is likely a typo. in the problem statement, but I am still interested in solubility.

REPLY [2 votes]: Note that $(x-2)^2=3^{x+1}$ and on can easily show that this has a unique real solution in the interval $(0,1).$ 
For a closed form solution, lets make a substitution $3^a=x-2$ for some $a\in\mathbb{R}.$ Then $3^{2a}=3^{x+1}$ and by the injectivity of exponentials $x=2a-1.$ Hence $3^a=2a-3$ and therefore the solution would be $$a=\dfrac{3-W(-\frac{3\sqrt{3}}2\ln 3)}{\ln 3}.$$