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

Answer

To solve the problem, we need to follow these steps:

### Step 1: Solve the equation
We start with the equation given in the problem:
\[ 2(x^2 + 1) = 5x \]

Expanding this, we have:
\[ 2x^2 + 2 = 5x \]

Rearranging the equation gives us:
\[ 2x^2 - 5x + 2 = 0 \]

### Step 2: Factor or use the quadratic formula
Next, we can factor the quadratic equation or use the quadratic formula. Let's factor it:
The equation \( 2x^2 - 5x + 2 = 0 \) can be factored as:
\[ (2x - 1)(x - 2) = 0 \]

Setting each factor to zero gives us the solutions:
1. \( 2x - 1 = 0 \) → \( x = \frac{1}{2} \)
2. \( x - 2 = 0 \) → \( x = 2 \)

So, the roots are \( x = \frac{1}{2} \) and \( x = 2 \).

### Step 3: Find \( x - \frac{1}{x} \)
Now, we need to find \( x - \frac{1}{x} \) for both values of \( x \).

**For \( x = \frac{1}{2} \):**
\[ x - \frac{1}{x} = \frac{1}{2} - \frac{1}{\frac{1}{2}} = \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2} \]

**For \( x = 2 \):**
\[ x - \frac{1}{x} = 2 - \frac{1}{2} = 2 - 0.5 = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

So, the values for \( x - \frac{1}{x} \) are:
- For \( x = \frac{1}{2} \): \( -\frac{3}{2} \)
- For \( x = 2 \): \( \frac{3}{2} \)

### Step 4: Find \( x^3 - \frac{1}{x^3} \)
Next, we calculate \( x^3 - \frac{1}{x^3} \) using the identity:
\[ x^3 - \frac{1}{x^3} = \left( x - \frac{1}{x} \right) \left( x^2 + 1 + \frac{1}{x^2} \right) \]

First, we need to find \( x^2 + \frac{1}{x^2} \):
Using the identity:
\[ x^2 + \frac{1}{x^2} = \left( x - \frac{1}{x} \right)^2 + 2 \]

**For \( x = \frac{1}{2} \):**
\[ x - \frac{1}{x} = -\frac{3}{2} \]
\[ x^2 + \frac{1}{x^2} = \left(-\frac{3}{2}\right)^2 + 2 = \frac{9}{4} + 2 = \frac{9}{4} + \frac{8}{4} = \frac{17}{4} \]
Now, substituting back:
\[ x^3 - \frac{1}{x^3} = \left(-\frac{3}{2}\right) \left(\frac{17}{4}\right) = -\frac{51}{8} \]

**For \( x = 2 \):**
\[ x - \frac{1}{x} = \frac{3}{2} \]
\[ x^2 + \frac{1}{x^2} = \left(\frac{3}{2}\right)^2 + 2 = \frac{9}{4} + 2 = \frac{9}{4} + \frac{8}{4} = \frac{17}{4} \]
Now, substituting back:
\[ x^3 - \frac{1}{x^3} = \left(\frac{3}{2}\right) \left(\frac{17}{4}\right) = \frac{51}{8} \]

### Final Answers
1. \( x - \frac{1}{x} = -\frac{3}{2} \) or \( \frac{3}{2} \)
2. \( x^3 - \frac{1}{x^3} = -\frac{51}{8} \) or \( \frac{51}{8} \)