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 sqrt%282sqrt%2810%29%2B11%29=x%2Bsqrt%28x%2By%29, find x-y.
CC13F #6
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
Given that x and y are integers and sqrt%282sqrt%2810%29%2B11%29=x%2Bsqrt%28x%2By%29, find x-y.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Your starting equation is  

    sqrt%282sqrt%2810%29%2B11%29 = x%2Bsqrt%28x%2By%29.


Square both sides

    2%2Asqrt%2810%29 + 11 = x%5E2 + 2x%2Asqrt%28x%2By%29 + %28x%2By%29.


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

    sqrt%2810%29 = x%2Asqrt%28x%2By%29,

    11 = x%5E2 + x + y.



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  sqrt%282sqrt%2810%29%2B11%29 = use your calculator = 4.16227766  (approximately).

              Right side  x%2Bsqrt%28x%2By%29 = 1+%2B+sqrt%281%2B9%29 = 1+%2B+sqrt%2810%29 = 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) About Me  (Show Source):
You can put this solution on YOUR website!


Consider the general expression

%28sqrt%28a%29%2Bsqrt%28b%29%29%5E2

Expanding this, we get

a%2B2sqrt%28ab%29%2Bb or %28a%2Bb%29%2B2sqrt%28ab%29

Turning this around, we have the general result

[*] sqrt%28%28a%2Bb%29%2B2sqrt%28ab%29%29=sqrt%28a%29%2Bsqrt%28b%29

In this problem, we have the expression

sqrt%2811%2B2sqrt%2810%29%29

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

sqrt%2811%2B2sqrt%2810%29%29=sqrt%281%29%2Bsqrt%2810%29=1%2Bsqrt%2810%29

The problem asks us to write that in the form

x%2Bsqrt%28x%2By%29

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