Positive real solutions for a and b in a+b+1/a+1/b+4=2(√(2a+1)+√(2b+1))

[anemone]

Determine all positive real $a$ and $b$ satisfying the equation $a+b+\dfrac{1}{a}+\dfrac{1}{b}+4=2(\sqrt{2a+1}+\sqrt{2b+1})$.

 

[anemone]

Solution of other:

Spoiler

Notice that $a+\dfrac{1}{a}+2-2\sqrt{2a+1}=\dfrac{a^2+2a+1-2a\sqrt{2a+1}}{a}=\dfrac{(a-\sqrt{2a+1})^2}{a}$ .

Hence the original equation can be rewritten as

$\dfrac{(a-\sqrt{2a+1})^2}{a}+\dfrac{(b-\sqrt{2b+1})^2}{b}=0$.

For $a,\,b>0$, this gives $a-\sqrt{2a+1}=0$ and $b-\sqrt{2b+1}=0$. It follows that the only solution is $a=b=1+\sqrt{2}$.