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

Answer

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

### Step 1: Simplify the equation
Start by expanding the left side of the equation:
\[ 2(x^2 + 1) = 5x \]
This simplifies to:
\[ 2x^2 + 2 = 5x \]

### Step 2: Rearrange the equation
Now, rearranging the equation to set it to zero:
\[ 2x^2 - 5x + 2 = 0 \]

### Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation \(2x^2 - 5x + 2\). We need to find two numbers that multiply to \(2 \times 2 = 4\) and add to \(-5\). The numbers \(-4\) and \(-1\) work:
\[ 2x^2 - 4x - x + 2 = 0 \]
Now, we can group the terms:
\[ (2x^2 - 4x) + (-x + 2) = 0 \]
Factoring out the common terms:
\[ 2x(x - 2) - 1(x - 2) = 0 \]
Now, factor out \((x - 2)\):
\[ (x - 2)(2x - 1) = 0 \]

### Step 4: Solve for \(x\)
Setting each factor to zero gives us:
1. \(x - 2 = 0 \Rightarrow x = 2\)
2. \(2x - 1 = 0 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}\)

### Step 5: Calculate \(x - \frac{1}{x}\)
Now we will calculate \(x - \frac{1}{x}\) for both values of \(x\).

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

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

### Final Result
Thus, the values of \(x - \frac{1}{x}\) are:
\[ \frac{3}{2} \text{ and } -\frac{3}{2} \]