Question

x2+1/x2=18,find x3-1/x3.

Answer 1

Given x^2 + 1/x^2 = 18.

          x^2 + 1/x^2 = 16 + 2

         x^2 + 1/x^2 - 2 = 16

         (x - 1/x)^2 = 16

        (x - 1/x) = 4.   ---- (1)


We know that a^3 - b^3 = (a-b)(a^2 + ab + b^2).

x^3 - 1/x^3 = (x - 1/x)(x^2 + x * 1/x + 1/x^2)

                  = (x - 1/x)(x^2 + 1 + 1/x^2)

                 = 4 * (18 + 1) (From (1))

                 = 4 * 19

                 = 76.


Hope this helps! :)

Answer 2

x²+1/x² = 18
(x-1/x)² + 2(x)(1/x) = 18
(x-1/x)² + 2 = 18
(x-1/x)² = 18 - 2
(x-1/x) = √16
(x-1/x) = ±4

Case (i)
x³ - 1/x³ = (x - 1/x)³ + 3(x)(1/x) (x - 1/x)
= (4)³ + 3(4)
= 64 + 12
= 76

Case (ii)
x³ - 1/x³ = (x - 1/x)³ + 3(x)(1/x) (x - 1/x)
= (-4)³ + 3(-4)
= -64 - 12
= -76

x³ - 1/x³ = 76 (or) -76

Hope it helps..!