This question has been really bugging me. Plotting the graph on a graphing calculator does give the values. Also, iterating the values of $x$ until and unless the equation gets satisfied could be another way. But instead of this approximation, I was wondering if there was a more analytical method to solving for the values of $x$.
$$x^2 - 4x + 4 = x\sin(x)$$
This is not an answer but it is too long for a comment.
Equations which mix polynomial and trigonomatric functions do not show analytical solutions (this is already the case for $x=\cos(x)$) and, just as BLAZE commented, only numerical methods (such as Newton) will allow to get solutions.
I cannot resist the pleasure of showing you how good approximations could be obtained using the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ which was proposed more than 1,400 years ago by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician (see here).
Using it, the equation becomes $$4 x^4-4 \pi x^3+\left(16+5 \pi ^2\right) x^2-\left(16 +20 \pi \right)\pi x+20 \pi ^2=0$$ which can be solved with radicals. The solutions being too long to be reported, I shall only give you the numerical solutions. For the quartic, the two real solutions are $$x_1\approx 1.047849$$ $$x_2\approx 2.874095$$ while, for the original equation, the solutions are $$x_1\approx 1.047481$$ $$x_2\approx 2.873069$$
One thing that's confusing me right now is that there are two solutions - the approximated values and the second set of numbers you have posted. Were those also an approximation using Newton-Rapson method? So, I guess, the values of x in this particular equation can only be approximated at best?
Also, an entirely different question. Wolfram Alpha gave me x1.04748 and x2.87307. Does it follow an entirely different computational process?
