Approximate Square Roots

We can find quite an accurate estimate of a square root using simple algebra. The square root of  
  is approximately 36 -  
, so the square root is just a little bit more.
We can find a better estimate by writing  
  is small.
  is small,  
  is very small and we can ignore it, so
\[1300 \simeq 1296+72x\]
\[4 \simeq 72x\]
\[x \simeq \frac{4}{72} =0.0555555...\]
\[\sqrt{1300} \simeq 36.055555...\]
We can improve this estimate by taking  
  into account. From (1) we have  
\[x^2+72x-4=0 \rightarrow x=\frac{4-x^2}{72}\]
We can use this as an iteration expression  
, and by putting  
  on the right hand side we get  
\[x_2=\frac{4-(1/18)^2}{72} \simeq 0.055513\]
, so that  
\[\sqrt{1300} \simeq 37.055513\]
This is correct to 8 significant figures.

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