talor series expansion of roots of algebraic equation

marellasunny

I have a algebraic equation like so:
x^2-1-εx=0

the roots are obviously-
x=ε/2±√(1+ε^2/4)

How can I expand the expression for the roots- as a taylor series?

the answer is given as:
x(1)=1+ε/2+ε^2/8+O(ε^3)

I am assuming the author expanded the root 'x' in terms of ε before hand and substituted in the algebraic.Is that even allowed?

mathman

Use the binomial theorem on √(1+ε^2/4) = (1 + ε²/4)^1/2 = 1+ε²/8 + O(ε⁴)