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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
 #3
avatar+1038 
+3

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
 #4
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+1

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

Guest Sep 1, 2020
 #5
avatar+111547 
+1

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
 #6
avatar+1038 
+2

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

ilorty  Sep 1, 2020
 #7
avatar+111547 
+2

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

Melody  Sep 1, 2020

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