How to show this formula to get a square root of a number in "just few seconds" is true? I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation.

Let $x$ be a non perfect square number. If $y$ is the closer perfect square to $x$ such that $y &lt; x$ then
$$\sqrt{x}\approx \sqrt{y}+\frac{x-y}{2\cdot \sqrt{y}}$$
And it gives at the maximum two correct decimals after the decimal point.

My first reaction to try to find from where this formula goes, was to expand it, I found $$2\sqrt{xy}\approx y +\sqrt{2y}\cdot(x-y)$$ and I tried to find a remarkable identity but I failed and I'm still stuck there. Also I don't know how should I prove the maximum of two correct digits after the comma.
If we look for example to the sqare root time of $1000$, $961$ is the closer perfect square which verifies the condition. Then we have $\sqrt{1000} \approx 31 + \frac{39}{2*31} = 31.62903...$ and with a calculator we have $\sqrt{1000} = 31.6227766017...$ which is quite good.
Any hints would be appreciate, thank you in advance.
To understand why this is a good approximation note that $(\sqrt{x})' = \frac{1}{2\sqrt{x}}$. 

So the formula is of the form $f(x) = f(y) + f'(y)(x-y)$. See Taylor's theorem for more details.
To find the square root of $x$ start with a guess of $y$, then  evaluate an updated value of $y$ using $(y+(x/y)) /2$. Repeat as many times as desired to get a more accurate answer to more decimal places. 
You may start with any value of $y$. Even 1 000 000 works.