Question

If x=1+√7,then the value of (-6)/x is​

Answer 1

Given, x - 1/x = 6 . . . . . . . . { 1 }

or (x - 1/x)² = 6²

or x² + 1/x² - 2 =36

or x² + 1/x² = 38

or x² + 1/x² + 2 = 40

or (x + 1/x)² = 40

so x + 1/x = ±2√10 . . . . . . . . { 2 }

Multiplying { 1 } and { 2 }

x² - 1/x² = ±12√10

or (x² - 1/x²)³ = (±12√10)³

or (x²)³ - (1/x²)³ - 3(x² - 1/x²) = ±17280√10

or (x²)³ - (1/x²)³ - 3(±12√10) = ±17280√10

or x^6 - 1/x^6 = ±17280√10 + (±36√10)

so x^6 - 1/x^6 = ±17316√10