Find explicitly the positive solutions of $2^x=x^2$ 
Find explicitly the positive solutions of the equation $2^x=x^2$

I noticed that $x=2$ and $x=4$ are roots of the equation.
How can I prove that they are the only positive ones? Thanks in advance

For positive $x$, we have $2^x=x^2\iff a^x=x$, where $a=\sqrt2$.  Since $0\lt a\not=1$, the exponential curve $y=a^x$ is concave up, hence any line intersects it at most twice. The line $y=x$ does so at $x=2$ and $x=4$.  So those are the only positive solutions of $2^x=x^2$. (As David Ullrich notes, there is also a negative solution; it corresponds to the interesection of the exponential curve $y=a^x$ and the line $y=-x$.)
