Question

Y= √a+2b + √a-2b ÷ √a+2b - √a-2b , prove that

by^{​​​​​2}​​​​-ay+b=0

Answer 1

The question only makes sense if the expression is written as

y=a+2b−−−−−√+a−2b−−−−−√a+2b−−−−−√−a−2b−−−−−√.y=a+2b+a−2ba+2b−a−2b.

We also assume that b≠0b≠0 (otherwise yy does not exist).

Multiplying yy by a+2b−−−−−√+a−2b−−−−−√a+2b−−−−−√+a−2b−−−−−√a+2b+a−2ba+2b+a−2b, we get

y=a+a2−4b2−−−−−−−√2by=a+a2−4b22b

from which

(2by−a)2=a2−4b2(2by−a)2=a2−4b2

and the result

by2−ay+b=0by2−ay+b=0