Question

if x3-1/x3=14,then x-1/x=

Answer 1

Let (x - 1/x) = p
Cubing on both sides of the equation,

(x - 1/x)3 = p3
x3 - 1/x3 - 3(x - 1/x) = p3
(14) - 3p = p3 [Given that x3 - 1/x3 = 14]
p3 + 3p - 14 = 0
(p - 2) is a factor of (p3 + 3p - 14) as p(2) = 0. [According to remainder theorem]
Therefore the other factor of (p3 + 3p - 14) on dividing by (p - 2) is (p2 + 2p + 7).
⇒ (p - 2)(p2 + 2p + 7) = 0
⇒ p = 2
The other values of p are not real numbers as (p2 + 2p + 7) can not be factored.
Therefore, (x - 1/x) = 2.

Answer 2

Answer:

x-1/x=2

Step-by-step explanation:

(x-1/x)³ = x³-1/x³ - 3×x×1/x(1–1/x)

or (x-1/x)³ = x³-1/x³ - 3(1–1/x)

or (x-1/x)³ = 14 - 3(1–1/x) . . . . [Given, x³-1/x³ = 14]

Let x-1/x = a

Therefore a³ = 14 - 3a

or a³+3a-14 =0

or a³-2a²+2a²-4a+7a-14 =0

or a²(a-2) +2a(a-2) +7(a-2) =0

or (a-2)(a²+2a+7) =0

or a-2 = 0/(a²+2a+7)

or a = 2

So x-1/x = 2