SOLUTION: If{{{x=sqrt(sqrt(a/b)-sqrt(b/a))}}} prove that {{{(2a)sqrt(1+x^2)/(x+sqrt(1+x^2))=a+b}}}

Algebra ->  Geometry-proofs -> SOLUTION: If{{{x=sqrt(sqrt(a/b)-sqrt(b/a))}}} prove that {{{(2a)sqrt(1+x^2)/(x+sqrt(1+x^2))=a+b}}}      Log On


   

Question 1132586: Ifx=sqrt%28sqrt%28a%2Fb%29-sqrt%28b%2Fa%29%29 prove that %282a%29sqrt%281%2Bx%5E2%29%2F%28x%2Bsqrt%281%2Bx%5E2%29%29=a%2Bb
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Sorry that's not true.

Here is a case where it's false:

suppose a=2, b=1, then

x=sqrt%28sqrt%28a%2Fb%29-sqrt%28b%2Fa%29%29 

x=sqrt%28sqrt%282%2F1%29-sqrt%281%2F2%29%29

Putting the right side in your calculator you get

x=0.8408964153

Then a=2, b=1 and x=0.8408964153

%282a%29sqrt%281%2Bx%5E2%29%2F%28x%2Bsqrt%281%2Bx%5E2%29%29=a%2Bb



Putting that in your calculator gives

2.433690671=3

which is clearly false.

Edwin

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

Thanks to Edwin, who disproved it and saved my time from spending for nothing.