I learnt it as the Newton-Raphson process, but whatever.

Basically, it is a way to estimate the root of an equation that is too complicated to solve easily.

f(x) represents the function of x, in this case being 4x³ - 3x - 1/2, and f'(x) is the differential of the function, in this case being 12x² - 3.

So, the equation is x - (4x³ - 3x - 1/2) ÷ (12x² - 3)

You use this by picking a starting value for x and substituting that into the above equation. The value it returns should be closer to the root than the value that you used. You can then use the new value to get an even better estimation until you have an answer to the required degree of accuracy.

So, if we start with x_0 = 1:

x_1 = 1 - (4 - 3 - 1/2) ÷ (12 - 3) = 0.9444...

Now we use this value and do it again.

x_2 = 0.9444... (4*(0.9444...)³ - 3*0.9444... - 12) ÷ (12*(0.9444...)² - 3)

= 0.9397...

x_3 = 0.9397...

Now that we have 2 values that are the same to 4 decimal places, we can stop and call that the answer. irspow used another iteration, but that was because he decided to be nice and give you an extra decimal place, free of charge.