Looking for solution of a 9th grader's problem

Question

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.

Answer 1

Don't over think it.   If you can't do it, do what can be done and give a reasonable explaination as to why more cannot be done.   If those reasons are correct, then you've correctly answered.

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I would mostly expect a nineth grader to manage the following by hand:$$\begin{align}3(x-2)^2+5 &= 3^{x+2}+5\\ (x-2)^2&= 3^{x+1}\tag{$\star$}\\ 2\ln \lvert x-2\rvert &=(x+1)\ln 3\\ x&= 2\log_3\lvert x-2\rvert-1\end{align}$$ In an exam, the iterative expresson is as close to a solution as they might get, unless they are given access to a calulator to furnish an approximation.

Once they reach $\star$ they should recognise that no tidy solution will exist; $x$ just won't be expressable as elementary functions of integers since it occurs as both a base and an exponent.   They should mention that intuition.

NB: Depending on the course curriculum, the student may have encountered the Lambert W function.   If so, that deserves attention.   Check to see if examples of this have been covered in class, and revise.

For bonus points a student should sketch a graph to verify that a real solution does infact exist, and where it approximately may.   A sugestion of about $0.15$ from a rough graph would be rather nice ($0.1$ to $0.2$ is a good range).

Answer 2

There is a recent publication which expresses the generalized lambert function as a taylor series:

https://arxiv.org/abs/1801.09904

I do not know if that qualifies as an analytic solution, however I think even the Lambert function can't be considered as such.

It does apply to the equation $x^2e^{ax}=c$.

Answer 3

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 $$a=\dfrac{3-W(-\frac{3\sqrt{3}}2\ln 3)}{\ln 3}.$$