SOLUTION: pls help me to answer this problem: Simplify: \sqrt{5+\sqrt{5+\sqrt{5+\ldots}}}

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Question 1106262: pls help me to answer this problem: Simplify: \sqrt{5+\sqrt{5+\sqrt{5+\ldots}}}
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
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Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Let x = .     (1)


It is what you ask to simplify.


Square both sides of (1). You will get

x^2 = 5 + x,     or

x^2 - x - 5 = 0.


Use the quadratic formula:

 =  = .


Since x, obviously, must be > 0, only positive root is the solution:  x = .


Answer.   = .

Solved !


======================
Amazingly, isn't it ?


Let me show you another, even more amazing calculation !


Simplfy .

Solution

Let x = .     (2)


It is what I want to simplify.


Square both sides of (2). You will get

x^2 = 2 + x,     or

x^2 - x - 2 = 0.


Use the quadratic formula:

 =  =  = .


Since x, obviously, must be > 0, only positive root is the solution:  x =  = 2.


Answer.   = 2.


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