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Question

If x+y=3 and xy=6, what is the value of the expression (x2+y2)2+4x2y2+4xy2+4x2y

A
36
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B
45
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C
81
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D
72
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Solution

The correct option is C 81
Given the expression
(x2+y2)2+4x2y2+4yx2+4xy2

On manipulating the terms of this expression we get,
(x2+y2)2+(2xy)2+(4xy)(x2+y2)

(x2+y2)2+(2xy)2+2×(2xy)(x2+y2)

On comparing with the identity (a+b)2=a2+b2+2ab we get,

(x2+y2)2+(2xy)2+2×(2xy)(x2+y2)=[(x2+y2)+2xy]2

Again applying the identity (a+b)2=a2+b2+2ab we get,
[(x2+y2)+2xy]2=[x2+y2+2xy]2=[(x+y)2]2=(x+y)4

Now, putting value of x+y=3 in the simplified equation we get,
(x+y)4=(3)4=81.


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