If x²- 5sqrt5 x + 1 = 0 , and x > 0, what is the value of (x³-1over x³) ?

If x²- 5sqrt5 x + 1 = 0 , and x > 0, what is the value of…

If $x^2- 5\sqrt5 x + 1 = 0$ , and x > 0, what is the value of $(x^3-{1\over x^3})$ ?

Show answer & explanation

Correct answer: B

Explanation

The correct answer is option 2 i.e. 1364.

(a - b)³ = a³ - b³ - 3ab(a - b)

Given, $x^2- 5\sqrt5 x + 1 = 0$

Dividing the given expression be x,

⇒ $x + { 1 \over x} = 5\sqrt{5}$

Now, $(x - { 1 \over x})^2 = (x + {1 \over x})^2 - 4$

⇒ $(x - {1 \over x})^2$ = 125 - 4

⇒ $x - {1 \over x}$ = 11

Now, cubing both the sides,

⇒ $(x - {1 \over x})^3$ = 1331

⇒ $x^3 - {1 \over x^3}$ - 3(11) = 1331

⇒ $x^3 - {1 \over x^3}$ = 1331 + 33

⇒ $x^3 - {1 \over x^3}$ = 1364