if x3-1/x3=14 find x-1/x
The formula says A3 - B3 = (A - B)(A2 + AB + B2), A2 + B2 = (A-B)2 +2AB
So, X3 - 1/X3=(X - 1/X)(X2 + X.1/X + 1/X2)=(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 X3 - 1/X3 = Y(Y2 + 3) = 14,
∴ Y3 + 3Y - 14 = 0
=> Y3 - 2Y2 + 2Y2 - 4Y + 7Y - 14 = 0
=> Y(Y - 2) + 2Y(Y - 2) + 7(Y - 2) = 0
=> (Y - 2)(Y2 + 2Y + 7) = 0
∴ Y = 2 = X - 1/X