Find all the solutions of $(a+1/b)(b+1/a)=4$

Question

I tried to find out some solutions to this equation but I found that it doesn't have any solution, but I couldn't find an elegant solution for this.

How are you going to find $a$ and $b$ from one equation ? – May 04 '17 at 01:18 — , May 04 '17 at 01:18

score 4 · Answer 1 · answered May 04 '17 at 01:16

4

Multiplying things out gives $$ ab+\frac1{ab}=2 $$ Which is $$ \frac{(ab-1)^2}{ab}=0 $$

answered May 04 '17 at 01:16

robjohn

score 2 · Answer 2 · answered May 04 '17 at 01:15

2

Well this simply boils down to:

$$(ab+1)^2=4ab$$ $$(ab-1)^2=0$$ $$ab=1$$

carry on forward from here

answered May 04 '17 at 01:15

The Dead Legend

score 1 · Answer 3 · answered May 04 '17 at 01:34

1

We have: $(a+1/b)(b+1/a) \ge 4$ by AM-GM inequality. Can you take it from here?

answered May 04 '17 at 01:34

DeepSea

3 Answers3