Let \(a,b,c,\) and \(d\) be distinct real numbers such that


\(\begin{align*} a &= \sqrt{4 + \sqrt{5 + a}}, \\ b &= \sqrt{4 - \sqrt{5 + b}}, \\ c &= \sqrt{4 + \sqrt{5 - c}}, \\ d &= \sqrt{4 - \sqrt{5 - d}}. \end{align*}\)



Compute \(abcd.\)

 Sep 1, 2020

For anyone who can't seem to see the LaTeX, here is the picture:



(I plugged it into a latex site and screenshoted it)


But I'm not just here to give you a picture of something you probably already know. 


After quite a long time of confusion, I finally got the answer! I actually first searched this question just to make sure that it wasn't answered before. And I found this link. Guest's answer confused me, because he/she said there was no answer. So I decided to try myself, and I got an answer to an "unanswerable" question. 


Let the variable 'x' represent a, b, c and d.


We can say this:



Square both sides:


Subtract 4:


Square again!


Multiply out the left side:


Subtract 5:


We could keep on going, but remeber, we want abcd, not the equation for it. Using vieta's formula, we see that abcd = 11/1 = 11


That is the answer to your "unanswerable" question wink



 Sep 1, 2020

Ohhhh I see this makes so much sense! Thank you so much!!!

Guest Sep 1, 2020

Thanks ilorty,

It took a little while for me to get my head around what you have done.

It is an impressively simple answer.    laugh cool

Melody  Sep 1, 2020

Thanks! Maybe I'll practice my explaination skills, so that it won't be that hard to understand my answers....

ilorty  Sep 1, 2020

It wasn't your explanation at fault, it was my brain.  LOL

Melody  Sep 1, 2020

12 Online Users