Question

please answer the question fast!!!!

Answer 1

Given Equation is x^2 + 1/x^2 = 17/4.

(1)

It can be written as:

⇒ x^2 + 1/x^2 = 9/4 + 2

⇒ x^2 + 1/x^2 - 2 = 9/4

⇒ (x - 1/x)^2 = 9/4

⇒ (x - 1/x) = 3/2    ------- (1)

(2)

It can be written as:

⇒ x^2 + 1/x^2 = 25/4 - 2

⇒ x^2 + 1/x^2 + 2 = 25/4

⇒ (x + 1/x)^2 = 25/4

⇒ x + 1/x = 5/2     ------ (2)

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On cubing (1) on both sides, we get

⇒ (x - 1/x)^3 = (3/2)^3

⇒ x^3 - 1/x^3 - 3(x - 1/x) = 27/8

⇒ x^3 - 1/x^3 - 3(3/2) = 27/8

⇒ x^3 - 1/x^3 - 9/2 = 27/8

⇒ x^3 - 1/x^3 = 27/8 + 9/2

⇒ x^3 - 1/x^3 = 63/8.      --------- (3)

On cubing (2) on both sides, we get

⇒ (x + 1/x)^3 = (5/2)^3

⇒ x^3 + 1/x^3 + 3(x + 1/x) = 125/8

⇒ x^3 + 1/x^3 + 3(5/2) = 125/8

⇒ x^3 + 1/x^3 + 15/2 = 125/8

⇒ x^3 + 1/x^3 = 125/8 - 15/2

⇒ x^3 + 1/x^3 = 65/8.

Hope it helps!