SOLUTION: x=sqrt(3+2sqrt(2)) + sqrt(3-2sqrt(2)) A) fund the exact value of x^2 B) hence, determine the simplified form of x

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Question 1118304: x=sqrt(3+2sqrt(2)) + sqrt(3-2sqrt(2))
A) fund the exact value of x^2
B) hence, determine the simplified form of x

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
x=sqrt(3+2sqrt(2)) + sqrt(3-2sqrt(2))
A) fund the exact value of x^2
B) hence, determine the simplified form of x
x=sqrt%283%2B2sqrt%282%29%29+%2B+sqrt%283-2sqrt%282%29%29

To make things easier, let

sqrt%283%2B2sqrt%282%29%29=A and sqrt%283-2sqrt%282%29%29=B

Then 

x=A+%2B+B

x%5E2+=+%28A%2BB%29%5E2

x%5E2+=+%28A%2BB%29%28A%2BB%29

x%5E2=A%5E2%2BAB%2BAB%2BB%5E2

x%5E2=A%5E2%2B2AB%2BB%5E2

Now since 

x%5E2+=+A%5E2%2B2AB%2BB%5E2, we'll need AČ, BČ, and AB

A%5E2+=+%28sqrt%283%2B2sqrt%282%29%29%29%5E2

A%5E2+=+3%2B2sqrt%282%29

B%5E2+=+%28sqrt%283-2sqrt%282%29%29%29%5E2

B%5E2+=+3-2sqrt%282%29

AB+=+%28sqrt%283%2B2sqrt%282%29%29%29%28sqrt%283-2sqrt%282%29%29%29%29%29

AB+=+sqrt%28%283%2B2sqrt%282%29%29%283-2sqrt%282%29%29%29

AB+=+sqrt%28%283%2B2sqrt%282%29%29%283-2sqrt%282%29%29%29

AB+=+sqrt%289-6sqrt%282%29%2B6sqrt%282%29-4%2A2%29%29

AB+=+sqrt%289-4%2A2%29

AB+=+sqrt%289-8%29

AB+=+sqrt%281%29

AB+=+1

Substituting in

x%5E2+=+A%5E2%2B2AB%2BB%5E2,

x%5E2+=+%283%2B2sqrt%282%29%29%2B2%281%29%2B%283-2sqrt%282%29%29,

x%5E2+=+3%2B2sqrt%282%29%2B2%2B3-2sqrt%282%29,

x%5E2+=+3%2B2%2B3,

x%5E2=8

x=%22%22+%2B-+sqrt%288%29

But x cannot be negative.

x=sqrt%284%2A2%29

x=sqrt%284%29sqrt%282%29

x=2sqrt%282%29

Edwin