A left to right algorithm for squaring 1.7 would look like
This illustrates the formula for squaring a binomial
The 2ab are the two .7 in blue in the computation.
If we use this technique to verify the next decimal place we get
In this case, we can let a = 1.7; because we know from the previous step that 1.72 = 2.89. If we let b = .03, then the 2ab are the two .051s that are in the blue box, and .032 = .0009.
We could get the two .051 by doubling the 1.7 and multiplying that by the .03. We could get everything that is added to the 2.89 by the following process.
When we add the 2.89 we get
This represents a tremendous streamlining in the guess and check process; because we can use the results of the previous step, and add in the results of the extra place of accuracy as simply as possible.
We are simplifying the formula for squaring a binomial
To check the square of 1.732, let a = 1.73 and b = .002. We know that 1.732 = 2.9929. Then to compute
When we add this to the 2.9929, we get