SOLUTION: If x^2+y^2=20 and xy=30, what is the value of (2x+2y)^2? Thank you in advance!

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Question 865791: If x^2+y^2=20 and xy=30, what is the value of (2x+2y)^2?
Thank you in advance!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If x^2+y^2=20 and xy=30, what is the value of (2x+2y)^2?


Thank you in advance!


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First factor out 2 and rewrite things a bit to get


%282x%2B2y%29%5E2+=+%282%28x%2By%29%29%5E2


%282x%2B2y%29%5E2+=+2%5E2%2A%28x%2By%29%5E2


%282x%2B2y%29%5E2+=+4%28x%2By%29%5E2


%282x%2B2y%29%5E2+=+4%28x%2By%29%5E2


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Now expand out %28x%2By%29%5E2 to get %28x%2By%29%5E2=x%5E2+%2B+2xy+%2B+y%5E2


So we have these 3 terms: x^2, 2xy, y^2


We have 2 of them so far. The outer terms x^2 and y^2 and they are from the equation x^2+y^2=20


We just need a 2xy term. Well we have xy=30, so let's multiply both sides by 2 to get 2xy=60.


That wraps up what we need. Let's put it all together.


From above, we found that %282x%2B2y%29%5E2+=+4%28x%2By%29%5E2. Replace %28x%2By%29%5E2 with x%5E2+%2B+2xy+%2B+y%5E2 (since %28x%2By%29%5E2=x%5E2+%2B+2xy+%2B+y%5E2) to get %282x%2B2y%29%5E2+=+4%28x%5E2+%2B+2xy+%2B+y%5E2%29


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Now group up the x^2 and y^2 terms


%282x%2B2y%29%5E2+=+4%28x%5E2+%2B+2xy+%2B+y%5E2%29


%282x%2B2y%29%5E2+=+4%28x%5E2%2By%5E2%2B2xy%29


%282x%2B2y%29%5E2+=+4%28%28x%5E2%2By%5E2%29%2B2xy%29


And replace x%5E2%2By%5E2 with 20 (because we're given x^2+y^2=20)


%282x%2B2y%29%5E2+=+4%28%2820%29%2B2xy%29


%282x%2B2y%29%5E2+=+4%2820%2B2xy%29


Then replace 2xy with 60 (since we just found that 2xy=60)


%282x%2B2y%29%5E2+=+4%2820%2B2xy%29


%282x%2B2y%29%5E2+=+4%2820%2B60%29


Now compute to finish up


%282x%2B2y%29%5E2+=+4%2820%2B60%29


%282x%2B2y%29%5E2+=+4%2880%29


%282x%2B2y%29%5E2+=+320