SOLUTION: 1/(x-1)^2+1/x-1=12

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Question 1132643: 1/(x-1)^2+1/x-1=12
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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1/(x-1)^2+1/x-1=12
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            See how I edited your post to make it  unambigous: 

              1/(x-1)^2 + 1/(x-1) = 12    <<<---=== I added parentheses !

Now the equation is presented  clearly,  and I am ready to solve it !


Solution 

Introduce new variable  z = 1%2F%28x-1%29.

Then your original equation takes the form


    z%5E2+%2B+z = 12,    or

    z%5E2+%2B+z+-+12 = 0.


Factor left side


    (z-3)*(z+4) = 0.


The two roots for z are  z= 3  and  z= -4.


Now we have two cases to analyse separately.


    Case 1.  z= 3  ====>  1%2F%28x-1%29 = 3  ====>  x-1 = 1%2F3  ====>  x= 11%2F3.


    Case 2.  z= -4  ====>  1%2F%28x-1%29 = -4  ====>  x-1 = -1%2F4  ====>  x= 3%2F4.


ANSWER.  There are two solutions  x= 11%2F3  and  x= 3%2F4.