If `2(x^(2) + 1)= 5x`, find
`x^(3)- (1)/(x^(3))`

Answer

To solve the equation \(2(x^2 + 1) = 5x\) and find \(x^3 - \frac{1}{x^3}\), we will follow these steps:

### Step 1: Simplify the Given Equation
Start with the equation:
\[ 2(x^2 + 1) = 5x \]
Divide both sides by 2:
\[ x^2 + 1 = \frac{5}{2}x \]
Rearranging gives:
\[ x^2 - \frac{5}{2}x + 1 = 0 \]

### Step 2: Solve for \(x + \frac{1}{x}\)
From the rearranged equation, we can express \(x + \frac{1}{x}\). We know:
\[ x + \frac{1}{x} = \frac{5}{2} \]

### Step 3: Find \(x - \frac{1}{x}\)
Using the identity:
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
We can find \(x^2 + \frac{1}{x^2}\):
\[ \left(\frac{5}{2}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Calculating gives:
\[ \frac{25}{4} = x^2 + 2 + \frac{1}{x^2} \]
Thus:
\[ x^2 + \frac{1}{x^2} = \frac{25}{4} - 2 = \frac{25}{4} - \frac{8}{4} = \frac{17}{4} \]

### Step 4: Find \(x - \frac{1}{x}\)
Using the identity:
\[ \left(x - \frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2} \]
We have:
\[ \left(x - \frac{1}{x}\right)^2 = \frac{17}{4} - 2 = \frac{17}{4} - \frac{8}{4} = \frac{9}{4} \]
Taking the square root gives:
\[ x - \frac{1}{x} = \pm \frac{3}{2} \]

### Step 5: Find \(x^3 - \frac{1}{x^3}\)
Using the identity:
\[ x^3 - \frac{1}{x^3} = \left(x + \frac{1}{x}\right)\left(x^2 - 1 + \frac{1}{x^2}\right) \]
We already know:
\[ x + \frac{1}{x} = \frac{5}{2} \]
And:
\[ x^2 - 1 + \frac{1}{x^2} = \frac{17}{4} - 1 = \frac{17}{4} - \frac{4}{4} = \frac{13}{4} \]
Thus:
\[ x^3 - \frac{1}{x^3} = \left(\frac{5}{2}\right)\left(\frac{13}{4}\right) = \frac{65}{8} \]

### Final Answer
The final result is:
\[ \boxed{\frac{65}{8}} \]