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.
#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:
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:
And that should be familar and suggest, if nothing else, Completing the Square or the Quadratic Formula.