Question

If ula">+$\frac{1}{x^{3}}$=18
Then prove, x=$\frac{3+\sqrt{5}}{2}$

Answer 1

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.

Answer 2

Answer:

x=  

2

3

​

+2

​

≈1.866025404  

Step-by-step explanation: