1. ## Help!

Can someone show me how to manipulate the following, in stages, to find x:

$\displaystyle x = \frac{4(\sqrt2 - 1) \pm \sqrt{(4(\sqrt2 - 1))^2 + 4(1)(29+8\sqrt2)}}{2}$

2. First, you can factor out 4 from the term in the large root:
$\displaystyle x=\frac{4(\sqrt{2}-1)\pm\sqrt{4(4(\sqrt{2}-1)^2+29+8\sqrt{2})}}{2}$
$\displaystyle x=\frac{4(\sqrt{2}-1)\pm{2}\sqrt{(4(\sqrt{2}-1)^2+29+8\sqrt{2})}}{2}$
$\displaystyle x=2(\sqrt{2}-1)\pm\sqrt{4(\sqrt{2}-1)^2+29+8\sqrt{2}}$.

Next, use that
$\displaystyle (\sqrt{2}-1)^2=(\sqrt{2})^2-2(1)\sqrt{2}+1^2$
$\displaystyle \;\;=2-2\sqrt{2}+1$
$\displaystyle \;\;=3-2\sqrt{2}$

So:
$\displaystyle x=2(\sqrt{2}-1)\pm\sqrt{4(3-2\sqrt{2})+29+8\sqrt{2}}$
$\displaystyle x=2(\sqrt{2}-1)\pm\sqrt{12-8\sqrt{2}+29+8\sqrt{2}}$
$\displaystyle x=2(\sqrt{2}-1)\pm\sqrt{41}$

-Kevin C.