Easy way to simplify $\frac{-1-2\sqrt{5}\pm\sqrt{21-4\sqrt{5}}}{4}$

Question

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!

Answer 1

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.

Answer 2

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