If `x + (1)/(x) = 5,` then the value of `(x)/(1 + x + x ^(2))` is

Options

A. `1/5`
B. `1/6`
C. `5`
D. 6

Answer

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

### Step 1: Solve for \( x \)
We start with the equation:
\[ x + \frac{1}{x} = 5 \]
To eliminate the fraction, we can multiply both sides by \( x \):
\[ x^2 + 1 = 5x \]
Rearranging gives us a quadratic equation:
\[ x^2 - 5x + 1 = 0 \]

### Step 2: Use the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 - 5x + 1 = 0 \), we have \( a = 1 \), \( b = -5 \), and \( c = 1 \). Plugging these values into the formula:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{5 \pm \sqrt{25 - 4}}{2} \]
\[ x = \frac{5 \pm \sqrt{21}}{2} \]

### Step 3: Find \( 1 + x + x^2 \)
Next, we need to find \( 1 + x + x^2 \). We already have \( x^2 \) from our quadratic equation:
\[ x^2 = 5x - 1 \]
Now substituting this into \( 1 + x + x^2 \):
\[ 1 + x + x^2 = 1 + x + (5x - 1) = 6x \]

### Step 4: Substitute back into the expression
Now we can substitute \( 1 + x + x^2 \) into our original expression:
\[ \frac{x}{1 + x + x^2} = \frac{x}{6x} = \frac{1}{6} \]

### Final Answer
Thus, the value of \( \frac{x}{1 + x + x^2} \) is:
\[ \frac{1}{6} \]