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This is my first question on StackExchange. So my question is:

If $$x = \frac{\sqrt{\sqrt5 +2}+\sqrt{\sqrt5-2}}{\sqrt{\sqrt5 + 1}} + \frac{\sqrt{\sqrt5 +2}+\sqrt{\sqrt5-2}}{2\sqrt{\sqrt5 + 1}} - \sqrt{3-2\sqrt2}$$ What is the value of $x^2$?

If someone can also tell me how to input mathematical functions on the StackExchange it would be appreciated.

Shahar
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arunkhanna00
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  • Welcome! Please edit this post and add some MathJax. – k170 Aug 27 '14 at 15:26
  • I edited your post, hope I didn't mess up. – Shahar Aug 27 '14 at 15:27
  • Tex goes between single or double dollar signs. \sqrt for square root, \frac or \dfrac for fractions. Anything longer than one character for formatted input goes in curly brackets{}. For example, \sqrt49 gives $\sqrt49$. – Mike Aug 27 '14 at 15:28
  • you can typeset many beautiful mathematical equations using MathJax on here :) – Hypergeometricx Aug 27 '14 at 15:31
  • @hypergeometric And some really ugly ones like the one above. :) It can be a lot of work sometimes, but sometimes neatness counts for a lot. – Mike Aug 27 '14 at 15:35
  • @Mike - the purpose is to reduce it to something much less ugly and more beautiful like the solutions given below! – Hypergeometricx Aug 27 '14 at 16:06

3 Answers3

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Letting $$\frac{\sqrt{\sqrt 5+2}+\sqrt{\sqrt 5-2}}{\sqrt{\sqrt 5+1}}=\alpha,$$ we have $$\alpha^2=\frac{(\sqrt 5+2)+(\sqrt 5-2)+2\sqrt{(\sqrt 5+2)(\sqrt 5-2)}}{\sqrt 5+1}=\frac{2(\sqrt 5+1)}{\sqrt 5+1}=2\Rightarrow \alpha=\sqrt 2.$$ Hence, we have $$x=\alpha+\frac{\alpha}{2}-\sqrt{(\sqrt 2-1)^2}=\frac{3}{2}\alpha-(\sqrt 2-1)=\frac{\sqrt 2}{2}+1\Rightarrow x^2=\frac{3}{2}+\sqrt 2.$$

mathlove
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I'd say the fastest way to do this is type it into a calculator (or some mathematical tool like matlab or R)

I might very likely have made a typo when entering your formula but my results are

  • $x\approx3.288246$
  • $x^2\approx10.81256$

please check it yourself!

Barkas
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  • Yeah, I know how to enter it in a calculator(using Google!) but I wanted to know how to solve it myself. Thanks for the reply though – arunkhanna00 Aug 28 '14 at 11:51
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Hint: $\sqrt{\sqrt5+2}\cdot\sqrt{\sqrt5-2}~=~?$, and $\sqrt{\sqrt5+1}\cdot\sqrt{\sqrt5-1}~=~?$ Also, if $(a-b)^2=a^2+b^2$

$-2ab=3-2\sqrt2$, then what are the values of a and b ?

Lucian
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