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) About Me  (Show Source):
You can put this solution on YOUR website!
What does the backslash mean?
what is ldots?

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let x = sqrt%285+%2B+sqrt%285+%2B+sqrt%285+%2B+ellipsis%29%29%29.     (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:

x%5B1%2C2%5D = %281+%2B-+sqrt%281%5E2+%2B+4%2A5%29%29%2F2 = %281+%2B-+sqrt%2821%29%29%2F2.


Since x, obviously, must be > 0, only positive root is the solution:  x = %281+%2B+sqrt%2821%29%29%2F2.


Answer.  sqrt%285+%2B+sqrt%285+%2B+sqrt%285+%2B+ellipsis%29%29%29 = %281+%2B+sqrt%2821%29%29%2F2.

Solved !


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Amazingly, isn't it ?


Let me show you another, even more amazing calculation !


Simplfy sqrt%282+%2B+sqrt%282+%2B+sqrt%282+%2B+ellipsis%29%29%29.

Solution 

Let x = sqrt%282+%2B+sqrt%282+%2B+sqrt%282+%2B+ellipsis%29%29%29.     (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:

x%5B1%2C2%5D = %281+%2B-+sqrt%281%5E2+%2B+4%2A2%29%29%2F2 = %281+%2B-+sqrt%289%29%29%2F2 = %281+%2B-+3%29%2F2.


Since x, obviously, must be > 0, only positive root is the solution:  x = %281+%2B+3%29%2F2 = 2.


Answer.  sqrt%282+%2B+sqrt%282+%2B+sqrt%282+%2B+ellipsis%29%29%29 = 2.


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