x/6 = 6/x-8 Answer is x = 4 + 2 times the square root of 13, but I don't know how to derive it

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- Jan 22nd 2009, 05:37 PMChuckShow how to solve for x
x/6 = 6/x-8 Answer is x = 4 + 2 times the square root of 13, but I don't know how to derive it

- Jan 22nd 2009, 05:59 PMLast_Singularity
- Jan 22nd 2009, 07:09 PMTKHunny
Gack! "Last" is sleeping on the job.

1) Please don't EVER casually multiply by 'x' wihtout at least considering the consequences. If x = 0, you are in big trouble!

2) Please write more carefully. Remember your Order of Operations. You have not written what you intended. You meant this:

x/6 = 6/(x-8)

Look at this and what you wrote and see the difference.

$\displaystyle

\frac{x}{6} = \frac{6}{x-8}

$

#1 Notice that x = 8 is no good. Why? No matter what else happens, do NOT believe that x = 8.

Having said that, multiply both sides by 'x-8'. This gives:

$\displaystyle

\frac{x \cdot (x-8)}{6} = 6

$

Multiply both sides by 6. Why was there no additional consideration this time? We know already that 6 is NOT zero (0).

Now we have:

$\displaystyle

x \cdot (x-8) = 36

$

or

$\displaystyle

x^{2}-8x = 36

$

or

$\displaystyle

x^{2}-8x - 36 = 0

$

And that should be familar and suggest, if nothing else, Completing the Square or the Quadratic Formula. - Jan 22nd 2009, 07:16 PMLast_Singularity
That's what the OP provided. I interpreted it as $\displaystyle x/6 = 6/x - 8$. If he meant $\displaystyle x/6 = 6/(x-8)$, he would have put the parenthesis there.

And from that, it is implied that $\displaystyle x \neq 0$. Otherwise the $\displaystyle 6/x$ would be undefined.

Whatever. - Jan 22nd 2009, 07:24 PMJhevon