# If #(5-sqrt(x))^2 = y-20sqrt(2)# where #x, y# are integers, then what are #x# and #y# ?

##### 1 Answer

#### Answer:

Expand and equate the irrational parts to help simplify and find:

#x = 8# and#y = 33#

#### Explanation:

Expand the left hand side:

#(5 - sqrt(x))^2#

#= 5^2-(2xx5xxsqrt(x))+x#

#= 25 - 10sqrt(x)+x#

#=(x+25)-10 sqrt(x)#

In order to eliminate the irrational

#-10 sqrt(x) = -20 sqrt(2)#

Divide both sides by

#sqrt(x) = 2sqrt(2)#

So

#x = (2sqrt(2))^2 = 8#

#y = x+25 = 8+25 = 33#

**Footnote**

Why is it possible to equate the irrational parts like this?

Consider the set of all numbers of the form

These representations are unique:

Suppose

Then

So if

So we must have