I was hoping to stay up until you worked all the steps out yourself, but I'm getting sleepy, so I'll just say what I have on my mind. I hope you work through each part rather than just reading it all at once.
The thing I hope you saw was to try \Delta y = 1, so that you'd get the relatively simple expression \sqrt{15 + 1} = \sqrt{15} + \Delta x, which is unfortunately tainted by the "complex" expression 1 \approx 2 \sqrt{15} \Delta x and doesn't simplify well.
Then, the thing I hoped you'd see next is that you might try y + \Delta y = 15. But now what else to set?
Either by chance, or by thoughtfully avoiding the problems with the previous attempt, I hope you'd try y = 16.
The main thing I was trying to demonstrate is that you often solve problems simply by blazing forward! Most of the steps were mostly your ideas -- you just had to be prompted to go forward. If you learn how to prompt yourself, then you'll be able to do a lot.
When you reach an obstacle, you then study it to see if the obstacle can be avoided. Or just try slightly tweaking things until you get around the obstacle. I like doing that a lot -- it often works quickly and without having to put a lot of thought into it.
(sometimes, our line of attack really doesn't work and we have to go back to square 1. But that didn't happen here)
Anyways, now that you have the final answer, can you find a shorter and clearer route to get there?
But don't despair! We don't always get things exactly right on the first try -- but often we still learn something that will be useful. With the equations you've worked out, is there something else you might try setting to 15?
. Good to know anyways though!
