It's very hard to solve, 2 is indeed a solution by observation but I think the best way to determine the other solutions is by using the graph of the 2 functions.
In case of x²=2^x the roots are :
x = 2 (obvious)
x = -0.766664695962123..
In the general case, the roots cannot be expressed in form of a combination of a finit number of elementary functions. Analytically, the solutions are expressed thanks to a special function, the Lambert function W(X)
x = -2 W(X)/ln(2) where X= -ln(2)/2
Since the Lambert function is multi-valuated two values are obtained for x.
More generally the roots of equation a(x^b)=c^x, with a,b,c constant parameters, are :
x = -b W(X)/ln(c) where X = - ln(c) / (b (a^(1/b)))
Depending on the values of the parameters, there are two, or one, or no solution(s).
Deciding where to start looking can be a bit of a trick. In this case, reasonable results can be obtained by a quick linearization or maybe a quadratic, since x^2 is quadratic already.
Produces x = 1.405 and x = -0.712
If we move up to the quadratic...
We get x = 1.691 and x = -0.778
Moving up to the cubic is a WHOLE LOT MORE WORK, but does manage an approximation of all three solutions,
x = 1.871, x = -0.765, and x = 12.582
That third one isn't pretty, but it could lead to something.