SOLUTION: if x^2 -16√(x) =12 then x-2√(x) = ??

Let me introduce new variable  u = .


Then I can reformulate the problem in this EQUIVALENT way:


    If   = ,  then find  .

If   = ,   then

     = 0.    (1)


The left side can be factored:

     = .


into the product of two quadratic polynomials.  (You may check the validity of this decomposition on your own).


Thus the equation (1) is equivalent to

    . = 0.     (2)


If "u" is the real root of the equation (1),  then "u" is the real root of the equation (2).

But the second polynomial (second multiplier)    is positively defined quadratic function  ,  which has NO real roots.


Therefore, "u" is the root of the first trinomial  of (2),  i.e.

     = 0,


Then   =  =  = .


Now, if   = ,  then   =  =  = 2.


     If   = ,  then   =  =  = 2.

    
Thus in any case   = 2.


It is the answer to the problem question.


Answer.  If x is a real number such as   = ,  then   = 2.