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algebra

by vishal chugh » Fri Nov 18, 2011 11:38 pm
what is the value of the expression below??
[x^2 + y^2]^2 - [x^2 - y^2]^2 divided by xy... it is given that x= (5)^1/2- (3)^1/2; y = (5)^1/2+ (3)^1/2

1. 2
2. 4
3. 6
4. 8
5. 16
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by shankar.ashwin » Fri Nov 18, 2011 11:48 pm
[x^2 + y^2]^2 - [x^2 - y^2]^2 = (x^2+y^2+x^2-y^2) * (x^2+y^2-x^2+y^2)

= 2(x^2 + y^2)

= 2* [ (√5-√3)^2 + (√5+√3)^2 ]

= 2* [ 5+3+5+3] = 32.

xy = (√5 + √3)*(√5 - √3) = 5-3 =2

Therefore, 32/2 = 16 E IMO
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by vishal chugh » Sun Nov 20, 2011 7:58 pm
here use (a+b)^2 = a^2 +b^2+2ab;
so (a-b)^2 = a^2 +b^2-2ab;
hence (a+b)^2 + (a-b)^2 = 4ab;
here a= x^2; b= y^2;
so expression becomes (4x^2*y^2)/xy or 4xy or 4(5-3) =8
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by Anurag@Gurome » Sun Nov 20, 2011 10:55 pm
vishal chugh wrote:what is the value of the expression below??
[x^2 + y^2]^2 - [x^2 - y^2]^2 divided by xy... it is given that x= (5)^1/2- (3)^1/2; y = (5)^1/2+ (3)^1/2

1. 2
2. 4
3. 6
4. 8
5. 16

Solution:
(a+b)^2 - (a-b)^2 = 4ab.
So, (x^2 + y^2)^2 - (x^2 - y^2)^2 = 4*x^2*y^2.
This divided by xy gives us 4xy = 4[5^(1/2) - 3^(1/2)][5^(1/2) + 3^(1/2)] = 4(5-3) = 8(using the identity that (a+b)(a-b) = a^2 - b^2).

The correct answer is option 4.
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