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If x + sqrt(xy) + y = 9 and x^2 + xy + y^2 = 27, find x - sqrt(xy) + y.

 Nov 22, 2019
 #1
avatar+105476 
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If x + sqrt(xy) + y = 9 and x^2 + xy + y^2 = 27, find x - sqrt(xy) + y.

 

x + sqrt (xy) + y  =  9

 

x + y  =  9 - sqrt (xy)           square both sides

 

x^2 + 2xy + y^2  =  81 - 18sqrt (xy) + xy     simplify

 

x^2 +xy + y^2  =  81 -  18 sqrt (xy)

 

27  =  81 - 81sqrt (xy)

 

-54  = -18 sqrt (xy)

 

3  =  sqrt (xy)

 

And

 

x + sqrt (xy) + y  =  9

 

x + (3 + y)  = 9

 

x + y  = 6

 

 

So

 

x - sqrt (xy) + y  =

 

x + y - sqrt (xy)  =

 

6   - 3   =   

 

3

 

 

cool cool cool

 Nov 23, 2019
 #2
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Notice that

 

\(\displaystyle (x+\sqrt{xy}+y)(x-\sqrt{xy}+y)=x^{2}+xy+y^{2},\\ \text{ so }\\9(x-\sqrt{xy}+y)=27,\\ \text{and therefore}\\ x-\sqrt{xy}+y = 3.\)

.
 Nov 23, 2019

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