Answer:
If x^3+1\x^3=18, then what is the value of x?
x^3 +1/x^3= 18.
Formula:-
(x +1/x)^3 = x^3. +1/x^3 + 3.(x +1/x).
Let (x+1/x)= p.
or, p^3 =18 +3p.
or, p^3 -3.p -18 =0.
Putting p=3 , remainder = 3^3–9–18 = 0 , (p-3) is a factor.
or, p^2.(p-3)+3.p^2-3p-18=0
or, p^2.(p-3)+3.p.(p-3)+6.(p-3)=0.
or, (p-3).(p^2+3p+6)=0
Either p-3=0.
or, p = 3. , putting p = x +1/x.
or, x +1/x = 3. ……………….(1).
And, (x -1/x)^2 = (x+1/x)^2 - 4. = 3^2 - 4 = 5.
Thus, x -1/x = +/-√5. …………….(2).
Adding eqn.(1) and (2).
2.x = 3+/-√5.
or, x = (3+/-√5)/2. Answer.
