SOLUTION: Given that x and y are integers and {{{sqrt(2sqrt(10)+11)=x+sqrt(x+y)}}}, find x-y. CC13F #6

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Question 1209471: Given that x and y are integers and , find x-y.
CC13F #6
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52818)   (Show Source): You can put this solution on YOUR website!
.
Given that x and y are integers and , find x-y.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Your starting equation is  

     = .


Square both sides

     + 11 =  +  + .


Now we want to follow the problem's instruction and find the solution 
in integer numbers x and y. For it, we combine irrationalities containing square roots 
in one equation and combine all the rest in the other equation.
Doing this way, you will get these two equations to find two unknown integer x and y

     = ,

    11 =  +  + .



Then trial and error will lead you to the unique solution  x= 1, y= 9  in integer numbers.


It gives then  x - y = 1 - 9 = -8.


ANSWER.  x - y = -8.


Final CHECK:  Left side   = use your calculator = 4.16227766  (approximately).

              Right side   =  =  = use your calculator = 4.16227766.


              * * *  All the digits coincide - hence, the solution is PRECISELY CORRECT  !  * * *

Solved.

----------------------

Notice that if the problem does not require x and y to be integer numbers,
then the given equation would have infinitely many solutions for two unknowns x and y in real numbers.

Thus, the requirement for the solutions of this equation to be integer numbers
extracts/identifies a unique solution among an infinite number of other possible real solutions.


                This is of  EXTREME  IMPORTANCE  ( ! )



Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Consider the general expression



Expanding this, we get

or

Turning this around, we have the general result

[*]

In this problem, we have the expression



Comparing that to the general result[*], we can see by inspection that



The problem asks us to write that in the form



So we want to have

x = 1 and x+y = 10

which leads us immediately to

x = 1 and y = 9.

Finally, the question asks for the value of x-y.

ANSWER: x-y = 1-9 = -8
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