$$x^2-4x-2^x=0$$
$$As\:\:x\rightarrow+\inf \:\:(x^2-4x-2^x)\rightarrow-\inf\\
As\:\:x\rightarrow-\inf \:\:(x^2-4x-2^x)\rightarrow+\inf$$
Therefore there must be an odd number of roots. I'm going to look for just 1
Finding the root of this is not super easy. It can't be factorised.
Let's consider $$f(x)x^2-4x-2^x$$
first I notice that as
$$If \:\: x=0 \mbox{ then } f(x)=0-0-1=-1$$
$$If \:\: x=-1 \mbox{ then } f(x)=1+4-1/2=+4.5$$
So there is at least one root between x=-1 ans x=0
Now use Newton's method of approximating roots to find it.
http://en.wikipedia.org/wiki/Newton's_method
There are you tube videos covering this as well. If you google them.
This is what the graph looks like
