# Thread: problem solving - square roots

1. ## problem solving - square roots

This problem was in my calculus exercise, yet I don't understand why:

Solve the equation :
$\displaystyle x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}$
(Show there are not other solutions to the ones you found)

Now, solving it regularly using simple algebra got me into an eighth-degree polynomial, which reminded my my teachers have just told us there are proofs for why there are no 'formulas' for solving polynomial equations higher than 5.

Anyway, how can I solve this problem using calculus tools? I thought of doing the following thing:
We know that $\displaystyle x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}$, therefore:
$\displaystyle x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}=\sqrt{5+\sqrt{5+\sq rt{5+\sqrt{5+\sqrt{5+\sqrt{5+x}}}}}}$.

Let the first expression be $\displaystyle a_3$, ane the second expression be $\displaystyle a_6$. Therefore:

$\displaystyle x=a_{3n}=\sqrt{5+\sqrt{5+\sqrt{5+...}}}$
Therefore,
$\displaystyle lim(x)=lim(a_{3n})$

but how do I find it ?

2. Does anyone have any idea ?

This looks like a nice one...

3. Well, if you think about it you can write:

$\displaystyle x = \sqrt{5+x}$ (can you understand why?)
So all you need to do is solve the quadratic equation you get after squaring both sides.

4. Originally Posted by Defunkt
Well, if you think about it you can write:

$\displaystyle x = \sqrt{5+x}$ (can you understand why?)
So all you need to do is solve the quadratic equation you get after squaring both sides.
Hmm.. I can only write $\displaystyle x = \sqrt{5+x}$, then try to see if it works and get back again $\displaystyle x = \sqrt{5+x}$. I don't think this is how you got to it, and how I should understand it.

This means it is a solution, but how can I know it's the only one?

5. $\displaystyle x = \sqrt{5+\sqrt{5+\sqrt{5+x}}} \Rightarrow x = \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+x} }}}}}$ $\displaystyle \Rightarrow x = \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+ \sqrt{...}}}}}}}$

So essentially, this is equivalent to $\displaystyle x= \sqrt{5+x}$, thus the solutions of the equation are the only ones which satisfy the condition for x.

6. hmm, okay, that makes sense, but how can I be sure there is no other solution?