If x = 2 + √3, find the value of (i) x + 1/x (
ii) x – 1/x
(iii) x^2 + 1/x^2
(iv) x^2 – 1/x^2
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Given that
x = 2 + √3
1/x = 1/2 + √3
= 1 × (2 - √3)/(2 + √3) (2 - √3)
= (2 - √3)/(2^2 - √3^2)
= (2 - √3)/4 - 3
= (2 - √3)
Therefore ,
x^2 = (2 + √3)
= (2)^2 + (√3)^2 + 2 × 2 × √3
= 4 + 3 + 4√3
= 7 + 4√3
1/x^2 = (2 - √3)^2
= (2)^2 + (√3)^2 - 2 × 2 × √3
= 4 + 3 - 4√3
= 7 - 4√3
x^2 + 1/x^2
= (7 + 4√3) + (7 - 4√3
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