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I'm teaching myself algebra 2 and I'm at this expression (I'm trying to find the roots):

$$ x=\frac{-1-2\sqrt{5}\pm\sqrt{21-4\sqrt{5}}}{4} $$

My calculator gives $ -\sqrt{5} $ and $ -\frac12 $ and I'm wondering how I would go about simplifying this down without a calculator. Is there a relatively painless way to do this that I'm missing? Or is it best to just leave these to the calculator? Thanks!

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up vote 3 down vote accepted

You'll want to try to write $21-4\sqrt 5$ as the square of some number $a+b\sqrt 5$. In particular, $$(a+b\sqrt 5)^2=a^2+2ab\sqrt 5+5b^2,$$ so we'll need $ab=-2$ and $21=a^2+5b^2$. Some quick trial and error shows us that $a=1,b=-2$ does the job.

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Short and sweet. Thank you! – dserver Mar 30 '13 at 13:58

$$21-4 \sqrt{5} = (1-2 \sqrt{5})^2$$

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And, just to be sure the OP doesn't overlook this, $\sqrt{(1-2\sqrt{5})^2}=|1-2\sqrt{5}|=2\sqrt{5}-1$ and not just $1-2\sqrt{5}$. – rschwieb Mar 30 '13 at 13:44
@rschwieb: right, but at some point, I wanted to leave something to the OP. – Ron Gordon Mar 30 '13 at 13:46

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