I saw on another post the question:
Given the expansion $(4 – 5x)^5 = 2 + (5/4)x + (25/64)x^2$, use $x = 1/10$ to find an approximation of $\sqrt{2}$. The answer is given there, but I'm struggling on how to even approach this problem, even with the discussion on the other thread. Any help would be much appreciated.
Best Answer
You still have the wrong sign in the first expression. Correcting that gives you $4/9$ instead of $20/35$ which has square root $2/3$ (the numbers were chosen to make this rational). Feeding in $x = 1/10$ now gives $181/128$, as claimed in the linked post. You can check that that is an approximation of root 2 by calculating its square.
If you want you know why this works, you could start with https://en.wikipedia.org/wiki/Binomial_approximation , particularly the generalisation near the end.