I want to solve the equation in $\mathbb{R}$: $$ \left|3^x - x\right|\left|3^x + x - 4\right| = 49 $$
My attempt:
The above equation is the same as: $$ \left(3^x - x\right)\left(3^x + x - 4\right) = \pm 49 $$
Case 1: $\left(3^x - x\right)\left(3^x + x - 4\right) = 49$
By taking derivative of the LHS, I managed to prove that it is monotonically increasing. Therefore, the equation has at most $1$ solution. By plugging in $\space x = 2,\space$ the equation is satisfied, so the first solution is $\space x = 2$.
Case 2: $\left(3^x - x\right)\left(3^x + x - 4\right) = -49$
I was not able to guess any integer nor rational solutions to this case, so I have to expand it: $$ x^2 - 4x + 4 \times 3^x - 3^{2x} = 49 $$ $$ (x - 2)^2 - \left(3^x - 2\right)^2 = 49 $$ This is where I got stuck and could not go further for a long time. I plugged it in WolframAlpha and received one solution: $\space x \approx -5.27928$.
I would like to know how to solve this equation appropriately, thanks in advance.
Even with special functions, you could not get an explicit solution for the zero of function $$f(x)=(x - 2)^2 - \left(3^x - 2\right)^2 - 49$$ and you will need some numerical method.
If you plot the function for $-6 \leq x \leq -5$, you will have a good surprise since, over that range $f(x)$ is very linear and this is very good for Newton method for example (or for any root-finding procedure). This is normal because $3^{-5}=\frac{1}{243} \ll 2$. So, for the time being, ignore that term and you will have $$x_0=2-\sqrt{53}\approx -5.28011$$
Making one single iteration of Newton method wiil give $$x_1=2-\sqrt{53}+\frac{9 \left(4\ 3^{\sqrt{53}}-9\right)}{2 \left(81 \log (3)+3^{\sqrt{53}} \left(3^{\sqrt{53}} \sqrt{53}-18 \log (3)\right)\right)}\approx -5.27928$$
Now, if you really want to polish the solution, continue ierating to get $$\left( \begin{array}{cc} n & x_n \\ 0 & \color{red}{-5.2}8010988928051827109730249152703279377766968257647743837818 \\ 1 & \color{red}{-5.2792786}8660406789534055303817937109818859725033462499045920 \\ 2 & \color{red}{-5.279278638758321}32633710054196086428743183092915950945577438 \\ 3 & \color{red}{-5.27927863875832116780469489398033}721735193951378723970745104 \\ 4 & \color{red}{-5.27927863875832116780469489398033547687818546758970688733708} \end{array} \right)$$
which, amuzing but with no meaning at all, is close to $$10\left(\sin \left(\frac{8 \pi }{59}\right)-\cos \left(\frac{9 \pi }{82}\right) \right)=-5.2792786221$$