If x + \(\frac{1}{x}\) = \(2\sqrt 5 \:\) where x > 1, then the value of x³- \(\frac{1}{{{x^3}}}\) is:

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 08 Dec 2022 Shift 4)

Answer 1

Given :

x + \(\frac{1}{x}\) = \(2√ 5 \:\) where x > 1,

Formula used:

(a + b)² = a² + b² + 2ab

(a - b)2 = a2 + b2 - 2ab

(a - b)³ = a³ - b³ - 3ab (a - b)

Calculation:

(x + \(\frac{1}{x}\) )² = x² + (\(\frac{1}{x}\) )² + 2

20 = x2 + (\(\frac{1}{x}\) )2 + 2

x2 + (\(\frac{1}{x}\) )2 = 20 - 2 =18 .....(1)

(x - \(\frac{1}{x}\) )2 = x2 + (\(\frac{1}{x}\) )2 - 2 , Put the value of equation (1),

(x - \(\frac{1}{x}\) )2 = 18 - 2 = 16

(x - \(\frac{1}{x}\) ) = 4 ....(2)

apply cube power to equation 2 both side,

=> (x - \(\frac{1}{x}\) )³ = 4³

=> (x - \(\frac{1}{x}\) )3 = x3 - \(\frac{1}{{{x^3}}}\) - 3( x - \(\frac{1}{x}\) ) = 64, put the value of equation (2)

=> x3 - \(\frac{1}{{{x^3}}}\) - 3( 4 ) = 64

=> x3 - \(\frac{1}{{{x^3}}}\) = 64 + 12

=> x3 - \(\frac{1}{{{x^3}}}\) = 76

Hence, '76' is the correct answer.

Shortcut Trick

If x + \(\frac{1}{x}\) = a then x - 1/x = √(a² - 4)

If x - 1/x = k then x³ - 1/x³ = k³ + 3k

Here, x + \(\frac{1}{x}\) = \(2\sqrt 5 \:\) then x - 1/x = √(20 - 4) = 4, then x3 - 1/x3 = (4³ + 12) = 76