This "relativistic kinetic energy" equation makes no sense to me

Presently, I'm reading an e-book I found on the internet titled "Relativity: The Special and General Theory", which may or may not have been written by Albert Einstein. Here's the part which has me in deep patatoes:

The author then mentions developing the equation into a series. I just can't understand how the second equation can represent kinetic energy.

Also, what's the difference between an equation and formula?

K=gmc^{2}-mc^{2}, then expand g in powers of v/c. The leading term in the expansion will be mc^{2}, which will cancel with the -mc^{2} in the expression for K. The surviving leading term will be (1/2)mv^{2}.

Keep reading and studying and it will start to make sense.

By using the binomial theorem, one can show that for normal, non-relativistic speeds--where v/c is small--that expression for relativistic KE is equivalent to the ordinary definition of 1/2mV^{2}. (That's what they mean by writing the equation as a series.) Here's a site that works it out: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html#c6

So the kinetic energy of an object would be the second equation minus mc^2? It gives good results when I test it. My calculator has a habit of rounding off numbers, how can i fix it?

It removes the rest energy, so, whatever is left over must be kinetic.

An equation relates two mathematical objects by declaring that they have the same value. It may or may not impose subordination of one object to another. A formula is a mathematical machine from which you put in your knowns to get a meaningful result. Subordination of the result is implied.