The quadratic x^2+10x+23=0 has roots -5+sqrt(2). We can write the quadratic x(x+10)=-23, so x=-23/(x+10). We can bring in sqrt(2) iteratively: put x=-5 into the expression on the right: x=-23/5=-4.6. Then we put x=-4.6 into the expression so we have x=-23/5.4=-4.259259... Put x=-4.259259 into the expression we get x=-4.0064516, and then we get x=-3.83746, and so on until after numerous iterations (repeating the process) we get to -3.585786438. This is very close to one of the roots of the equation, in other words -5+sqrt(2). The initial value of -5 guides the process to this root rather than the other root. Therefore, we can evaluate sqrt(2) by adding 5 to this number: 5-3.585786438=1.414213562. This iterative process can be performed by a calculator by repeatedly putting the value of x back into the same expression to get the next value of x. Some calculators allow you to build up an expression and keep applying it iteratively as you get each approximation. Using this method you can see the values converging until you reach the accuracy limits of the calculator and you get an unchanging result.
You can apply this method using x=1/(x-2). Start with x=1, then you get x=-0.5, then -0.4, then -0.4167 up to -0.4142135624. Then you add 1 to get sqrt(2). This gives you a more accurate answer because we have an extra digit 4 at the end. The iterative process can give you slightly better accuracy than simply working out sqrt(2) directly on your calculator.