Question

If x^2-3x+1=0,find the value of x^3-1/x^3

Answer 1

Answer:

8√5

Step-by-step explanation:

x² - 3x + 1 = 0

=> x² + 1 = 3x

=> ( x² + 1 ) / x = 3

=> x²/x + 1/x = 3

=> x + 1/x = 3

Since ( a - b )² = ( a + b )² - 4ab

=> ( x - 1/x )² = ( x + 1 / x )² - 4( x )( 1/x )

=> ( x - 1/x )² = 3² - 4

=> ( x - 1/x )² = 9 - 4

=> x - 1/x = √5

Cubing on both sides

=> ( x - 1/x )³ = (√5)³

Using ( a - b )³ = a³ - b³ - 3ab(a - b)

=> x³ - 1/x³ - 3( x )( 1/x )( x - 1/x ) = 5√5

=> x³ - 1/x³ - 3( √5 ) = 5√5

=> x³ - 1/x³ = 5√5 + 3√5 = 8√5