Approximation of square roots Recently, I've seen a YouTube video where they approximate square roots real quick.
They use this approximation :
$$\sqrt{x} \approx \lfloor \sqrt{x} \rfloor+\dfrac{x-(\lfloor \sqrt{x} \rfloor)^2}{2\lfloor \sqrt{x} \rfloor}$$
I want to know the math behind this approximation. Can someone help me out?

Let, $n \in N $
Suppose, we want to evaluate $\sqrt{n^2+c}$
$$\implies \sqrt{n^2(1+\dfrac{c}{n^2})}$$
$$\implies n \sqrt{1+\dfrac{c}{n^2}}$$
Now, we shall use the binomial expansion for $\sqrt{1+\dfrac{c}{n^2}}$
We know, $\sqrt{1+\dfrac{c}{n^2}}=1+\dfrac{c}{2n^2}-\dfrac{c^2}{8n^4}+\cdots$
So, $n \sqrt{1+\dfrac{c}{n^2}}=n+\dfrac{c}{2n}-\dfrac{c^2}{8n^3}+ \cdots$
Therefore, we can say, $\sqrt{n^2+c}=n+\dfrac{c}{2n}+\cdots$
From there we get the approximation, $\sqrt{x} \approx \lfloor x \rfloor + \dfrac{x-(\lfloor \sqrt{x} \rfloor)^2}{2\lfloor \sqrt{x} \rfloor}$
