For a more extended discussion of Vedic duplex method, with some background, and links to other aspects of Vedic Mathematics, click here. There is a follow-up lesson here, and additional links to go back and review, etc. Most of the sources, references, explanations, etc., that I found cited Wikipedia's definition, including the example given there. So, given your trouble understanding that entry, I thought I'd find something more directly from the source, with additional examples/steps, etc. It is very easy to follow, and begins by listing "observations"...then defines the "duplex" of a number...offering examples, and laying all the groundwork for the "vocabulary"/definitions of words used to describe the algorithm, before proceeding to actually taking on how this preliminary information can be used to find the square root of a number.
The duplex of a number is also called the Dvandva Yoga of a number. The duplex of a number is calculated as below:
For a single digit number, the duplex is simply the square of the number. Thus the duplex of 2 is 4, the duplex of 6 is 36 and so on
For a 2-digit number, the duplex is simply twice the product of the 2 digits of the number. Thus, the duplex of 16 is 2x1x6 = 12, the duplex of 90 is 2x9x0 = 0, the duplex of 43 is 2x4x3 = 24, and so on
For n-digit numbers, the duplex is calculated as the sum of several individual duplexes. Pair up the first digit with the nth digit of the number and find the duplex of the resulting 2-digit number. Similarly pair up the second digit with the (n-1)th digit and find the duplex of the resulting 2-digit number. Continue this process until no more 2-digit pairs can be formed. If a middle digit (that could not be paired with anything else exists at the end of the process, find its duplex also individually. Then add up all the duplexes found. The resulting sum of the duplex of the n-digit number
Some duplexes are shown below for illustration and to make sure the calculation of the duplex is fully understood.
$2: 2^2 = 4$
$5: 5^2 = 25$
$48: 2\times 4\times8 = 64$
$91: 2\times9\times1 = 18$
$314: 2\times3\times4 + 1^2 = 25$
$725: 2\times7\times5 + 2^2 = 74$
$350: 2\times3\times0 + 5^2 = 25$
$9357: 2\times9\times7 + 2\times3\times5 = 156$
$62787: 2\times6\times7 + 2\times2\times8 + 7^2 = 165$
Now that we know how to calculate the duplex of a number, how do we use it in calculating the square root of a number? We will explain using simple examples....
Why don't you read through the lessons (linked above), with paper and pencil, following along through examples, etc. Then, if you have specific questions about the method, or an example, feel free to edit your post with more specifics, or simply comment below my post, and I'll try to help clarify where you're stuck.