if x3-1/x3=14 find x-1/x

The formula says A^{3 }- B^{3} = (A - B)(A^{2 }+ AB + B^{2}), A^{2} + B^{2} = (A-B)^{2} +2AB

So, X^{3 }- 1/X^{3}=(X - 1/X)(X^{2 }+ X.1/X + 1/X^{2})=(X - 1/X)[(X - 1/X)^{2 }+ 2.X.1/X + X.1/X]=(X-1/X)[(X-1/X)^{2 }+ 3]

say, X-1/X=Y, then X^{3} - 1/X^{3} = Y(Y^{2} + 3) = 14,

∴ Y^{3 }+ 3Y - 14 = 0

=> Y^{3 }- 2Y^{2 }+ 2Y^{2} - 4Y + 7Y - 14 = 0

=> Y(Y - 2) + 2Y(Y - 2) + 7(Y - 2) = 0

=> (Y - 2)(Y^{2} + 2Y + 7) = 0

∴ Y = 2 = X - 1/X