What Values of k Make the Equation Have One Non-Negative Root?

[anemone]

Hi MHB, sorry for missing one week of high school's POTW, I guess I can make it up by posting two POTWs this week.(Blush)

Here is this week's another POTW:

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Find all values of $k$ such that the equation

$\left(\dfrac{1}{x+k}+\dfrac{k}{x-k}-\dfrac{2k}{k^2-x^2}\right)(|x-k|-k)=0$

has exactly one non-negative root.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to kaliprasad for his correct solution, which you can find below::)

Spoiler

Note that we cannot have $x = k$ or $x = -k$.

Multiply both sides by $ (x^2-k^2)$, we get

$((x-k) + k(x+k) + 2k) (\left| x -k \right| -k) = 0$

or $(x+k) (k+1) (\left| x -k \right| -k) = 0$

As $x$ cannot be $-k$ we have

$(\left | x -k \right| -k) = 0$

$|x-k | = k$, this gives 2 values of $x$, where $x = 0$ or $2k$.

Hence there is no solution to the problem.