Question

if x + 1/x = -1 ,
then ,
prove that
x³ = 1

Answer 1

X+1/X=-1
X^2+1=-X
X^2+X+1=0----(1)
WE KNOW THAT
(X-Y)^3=X^3-Y^3-3XY(X-Y)
(X-Y)^3+3XY(X-Y)=X^3-Y^3
(X-Y)((X-Y)^2+3XY)=X^3-Y^3
(X-Y)(X^2+Y^2-2XY+3XY)=(X^3-Y^3)
(X-Y)(X^2+Y^2+XY)=X^3-Y^3------(2)
sub x=X;y=1 in (2)
(x-1)(x^2+1+x)=x^3-1
from (1)
(x-1)(0)=x^3-1
0=x^3-1
x^3=1

Answer 2

Answer:

${x}^{3} = 1 \: \text{proved}$

Step-by-step explanation:

Given :

$x + \dfrac{1}{x} = - 1$

On further simplification we get :

${x}^{2} + 1 = - x \\ \\ {x}^{2} + 1 + x = 0$

We have identity;

${x}^{3} - {1}^{3} = (x - 1)( {x}^{2} + 1 + x)$

Putting above value here we get :

${x}^{3} - 1 = (x - 1) \times 0 \\ \\ {x}^{3} - 1 = 0 \\ \\ {x}^{3} = 1$

Hence proved.