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

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Question 1189763: Given that x and y are integers and +sqrt%28+2sqrt%28+10+%29+%2B+11+%29+=+x+%2B+sqrt%28+x+%2B+y+%29+, find x - y.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

                I just solved it yesterday.   See the link


https://www.algebra.com/algebra/homework/equations/Equations.faq.question.1189716.html

https://www.algebra.com/algebra/homework/equations/Equations.faq.question.1189716.html



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Consider the general expansion

%28a%2Bsqrt%28b%29%29%5E2=%28a%5E2%2Bb%29%2B2a%28sqrt%28b%29%29

You can use that pattern to attempt to find the value of an expression of the form

sqrt%28m%2Bsqrt%28n%29%29

In this problem, the expression under the radical on the left is

2%2Asqrt%2810%29%2B11 or 11%2B2%2Asqrt%2810%29

According to the pattern above, we should have a%5E2%2Bb=11, a=1, and b=10.

And those are all satisfied when a=1 and b=10. So

sqrt%2811%2B2%2Asqrt%2810%29%29+=+1%2Bsqrt%2810%29

Replacing the left side of the given equation with that gives us

1%2Bsqrt%2810%29=x%2Bsqrt%28x%2By%29

Solving by equating the rational and irrational parts of the two expressions gives us x=1 and y=9; so then

ANSWER: x-y = -8