Evaluate $\sqrt{41}$ to $n$ decimal places without using a calculator. Evaluate $\sqrt{41}$ to $n$ decimal places without using a calculator.
$$x^2 -10x-16=0$$
I was asked to solve the above quadratic giving the solutions to one decimal place.
Using the quadratic formula I got: $x= 5\pm \sqrt{41}$ 
Using a calculator that is: $11.4(1dp)$ or $-1.4(1dp)$
Then I wondered if I could solve this without a calculator.
It can be done by trial and error squaring values and gradually getting closer to the required accuracy. Here are the values that I squared to get to the required accuracy for the above case. $6.5,6.4,6.45,6.43,6.41$ and $6.405$. But is there another cleaner way of solving a surd to some number of decimal places?

Well, $42=6\cdot7$, so $6.5^2=42.25$, meaning $\sqrt{41}=\sqrt{42.25-1.25}\approx6.5-\frac{1.25}{2\cdot6.5}\approx6.4$. It's not really different from the other answer, the first step of Newton method, essentially, with $x_0=6.5$.
