If ` x + (1)/( x) = 3 ( x ne 0 )` then what is the value of ` (( x^(2) - x + 1))/( 5 x)` ?

Options

A. `(2)/( 5)`
B. `(3)/( 5)`
C. `(4)/( 5)`
D. `(1)/(5)`

Answer

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

### Step 1: Start with the given equation
We have:
\[ x + \frac{1}{x} = 3 \]

### Step 2: Multiply both sides by \( x \) to eliminate the fraction
Multiplying both sides by \( x \) (where \( x \neq 0 \)):
\[ x^2 + 1 = 3x \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ x^2 - 3x + 1 = 0 \]

### Step 4: Solve the quadratic equation using the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 - 3x + 1 = 0 \), we have \( a = 1 \), \( b = -3 \), and \( c = 1 \). Plugging these values into the formula:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{3 \pm \sqrt{9 - 4}}{2} \]
\[ x = \frac{3 \pm \sqrt{5}}{2} \]

### Step 5: Substitute \( x \) into the expression \( \frac{x^2 - x + 1}{5x} \)
Now we need to find the value of:
\[ \frac{x^2 - x + 1}{5x} \]
We can express \( x^2 \) in terms of \( x \) using the original equation:
\[ x^2 = 3x - 1 \]
Substituting this into the expression:
\[ \frac{(3x - 1) - x + 1}{5x} = \frac{3x - x - 1 + 1}{5x} = \frac{2x}{5x} \]

### Step 6: Simplify the expression
The \( x \) in the numerator and denominator cancels out:
\[ \frac{2x}{5x} = \frac{2}{5} \]

### Final Answer
Thus, the value of \( \frac{x^2 - x + 1}{5x} \) is:
\[ \frac{2}{5} \]