## 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
$\frac{36.0555555-36.0555127}{36.0555127} \times 100% = 1.2 \times 10^{-4} %=$
.