Estimating a Square Root With Algebra

We can find quite an accurate estimate of a square root using simple algebra. The square root of  
\[1300\]
  is approximately 36 -  
\[36^2=1296\]
, so the square root is just a little bit more.
We can find a better estimate by writing  
\[\sqrt{1300}=36+x\]
  where  
\[x\]
  is small.
Then  
\[1300=(36+x)^2=1296+72x+x^2\]
.
Since  
\[x\]
  is small,  
\[x^2\]
  is very small and we can ignore it, so
\[1300 \simeq 1296+72x\]
.
\[4 \simeq 72x\]
.
\[x \simeq \frac{4}{72} =0.0555555...\]
.
Then  
\[\sqrt{1300} \simeq 36.055555...\]
.
In fact  
\[\sqrt{1300}=36.0555127...\]
.
The error is  
\[\frac{36.0555555-36.0555127}{36.0555127} \times 100% = 1.2 \times 10^{-4} %=\]
.

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