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I am having trouble figuring this out.

$\sqrt {1+\left(\frac{x}{2}- \frac{1}{2x}\right)^2}$

I know that $\left(\frac{x}{2} - \frac{1}{2x}\right)^2=\frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2}$ but I have no idea how to factor this since I have two x terms with vastly different degrees, 2 and -2.

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    @Argon Make some browsing in his profile - specially, see the number of questions asked, the number of questions that are essentially the same problem with different variables, and how the OP doesn't put much of his effort in either solving the problem or understanding the hints or help of other users. [No offense intended to **anyone**.]2012-06-07

3 Answers 3

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Since $\begin{equation*} \left( \frac{x}{2}-\frac{1}{2x}\right) ^{2}=\frac{x^{2}}{4}-\frac{1}{2}+ \frac{1}{4x^{2}}, \end{equation*}$

we have

$\begin{eqnarray*} 1+\left( \frac{x}{2}-\frac{1}{2x}\right) ^{2} &=&1+\left( \frac{x^{2}}{4}- \frac{1}{2}+\frac{1}{4x^{2}}\right) \\ &=&1+\frac{x^{2}}{4}-\frac{1}{2}+\frac{1}{4x^{2}} \\ &=&\frac{x^{2}}{4}+\left( 1-\frac{1}{2}\right) +\frac{1}{4x^{2}} \\ &=&\frac{x^{2}}{4}+\frac{1}{2}+\frac{1}{4x^{2}} \\ &=&\left( \frac{x}{2}+\frac{1}{2x}\right) ^{2}, \end{eqnarray*}$

because

$\left( \frac{x}{2}+\frac{1}{2x}\right) ^{2}=\frac{x^{2}}{4}+\frac{1}{2}+ \frac{1}{4x^{2}}.$

Therefore $\begin{equation*} \sqrt{1+\left(\frac{x}{2}-\frac{1}{2x}\right)^2}=\sqrt{\left(\frac{x}{2} +\frac{1}{2x}\right)^2}=\left\vert\frac{x}{2}+\frac{1}{2x}\right\vert . \end{equation*}$

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Hint: for any (real, complex) numbers $\,a,b,\,$: $4ab+(a-b)^2=(a+b)^2$

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    @AméricoTavares That is fine but talking down to me like this is some basic thing all 12 years olds know and that my having no concept of it is a travesty is taking it too far.2012-06-07
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I presume you aren't asked to solve this (since it isn't an equation), but rather are asked to express it in a tidier form. Carrying on, we have \begin{align*} 1+\left(\frac{x^2}{4}-\frac{1}{2}+\frac{1}{4x^2}\right) &=\frac{x^2}{4}+\frac{1}{2}+\frac{1}{4x^2}\\\\ &= \left(\frac{x}{2}+\frac{1}{2x}\right)^2\\\\ &= \left(\frac{x^2+1}{2x}\right)^2 \end{align*}

Carry on from there: put the whole expression under the radical, use the fact that $\sqrt{a^2}=\mid a\mid$ to get $ \left|\frac{x^2+1}{2x}\right| $

By the way, this idiom, $4ab+(a-b)^2=(a+b)^2$, is very common and should eventually be part of your mathematical toolkit.