Question

If x=route5+2, then prove that x square 2 +1/x square 2=18.

Answer 1

x = √5 + 2
Now,
x^2 = (√5+2)^2 = 5+4+4√5 = 9+4√5.

And,
1/x^2 = 1/9+4√5
= 1/9+4√5 * 9-4√5/9-4√5
= 9-4√5 / (9)^2-(4√5)^2
= 9-4√5 / 1 = 9-4√5.

Now,
x^2 + 1/x^2 = 9+4√5 + 9-4√5 = 18.

Answer 2

Answer :

Given that,

x = √5 + 2

So, 1/x

= 1/(√5 + 2)

= (√5 - 2)/{(√5 + 2) (√5 - 2)}

= (√5 - 2)/(5 - 4)

= √5 - 2

Now, x + 1/x

= √5 + 2 + √5 - 2

= 2√5

So, x² + 1/x²

= (x + 1/x)² - 2 (x) (1/x)

= (2√5)² - 2

= 20 - 2

= 18

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