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Recently, a friend had asked me to find all values of $x$ which satisfy $9^{x^2 + 1} + 3^{x + 2} = 18$. My immediate thought was to divide both sides by $9$ to obtain $9^{x^2} + 3^{x} = 2$. The problem was initially a homework problem, and my friend had been able to deduce that $x = 0$ was a solution (noting that in this case, $9^{x^2} = 9^0 = 1$ and $3^{x} = 3^{0} = 1$, so their sum is $2$) prior to asking me. I had also found from this representation of the equation that $x$ is nonpositive; otherwise, both $9^{x^2}$ and $3^x$ are greater than $1$, so their sum is greater than $2$. This answer was accepted as correct by the people who had assigned the homework, but upon checking Wolfram Alpha we realized there appeared to be a second, negative root of the equation at around $x = -0.357941$. However, there was no exact form listed. Another friend attempted to use the $approxFraction$ function on a TI-nspire CX CAS, but the calculator was unable to find an approximation for it. We're suspecting that either the second solution is an incredibly complicated combination of logarithms, square roots, and fractions, OR there is no exact method to calculate the solutions, and the calculators simply used an approximation method (Taylor Series Polynomial/Newton's method). Could someone give us guidance on a possible solution method or show us why it may be impossible to deduce an exact solution method in a reasonable amount of time? Thanks!

Cogent Coder
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    For $x \in [-1,x_0),$ where $x_0$ is the negative solution, the function $f(x)=9^{x^2}+3^x-2$ satisfies $f(x)>0$ and $f'(x)<0$ and $f''(x)>0,$ so Newton's Method, starting with the crude approximate value $x=-1,$ will quickly lead to a good approximation. There is no formula for $x_0.$ – DanielWainfleet Nov 21 '21 at 07:54
  • I am pretty sure that the second solution can only be found numerically. But I have no proof. In most cases, Wolfram finds exact solutions, if possible. Since Wolfram alpha did not report an exact solution, this is another indication that there is none. – Peter Nov 21 '21 at 10:09
  • @DanielWainfleet is there a proof for the lack of a formula for $x_{0}$? I'd like to know more about this impossibility... – Cogent Coder Nov 24 '21 at 05:40
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    One of the first proofs of the lack of a conventional formula was for $\int e^{x^2}dx$ (Liouville). Such proofs can be difficult. I have learned that mixing logs or exponents with powers almost surely means no formula, but I don't really have a proof in this case. – DanielWainfleet Nov 25 '21 at 06:14

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