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
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.
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.