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Question

If √5+1/√5-1 y=√5-1/√5+1

find x2+y2

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Solution

x = root 5+ 1/root 5 - 1
rationalising it's denominator,
= √5+ 1/√5 - 1× √5 +1/ √5 + 1
= (√5 +1)2 / (√5)2 -(1)2
= 5+1 + 2 √5/ 5-1
= 6 + 2 √5/ 4
= 2(3 +√5)/4
= 3 +√5
.........
y = root 5 - 1/root 5 + 1
=
rationalising it's denominator,
= √5- 1/√5 + 1× √5 -1/ √5 - 1
= (√5 -1)2 / (√5)2 -(1)2
= 5+1 - 2 √5/ 5-1
= 6- 2 √5/ 4
= 2(3 -√5)/4
= 3 -√5


x²+y²= (3+√5)² + (3-√5)²

=(3²+2×3×√5+(√5)²)+(3²-2×3×√5+(√5)²)

=(9+6√5+5)+(9-6√5+5)
re arrange the terms
=9+9+5+5+6√5-6√5
x²+y²=28

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