If `x + (1)/(x) = 5 ` then the alue of `( 6x )/( x ^(2) + x +1 )` is,

Options

A. 1
B. 2
C. 3
D. 45

Answer

To solve the problem, we start with the equation given:

1. **Given Equation**:
\[ x + \frac{1}{x} = 5 \]

2. **Multiply both sides by \(x\)** to eliminate the fraction:
\[ x^2 + 1 = 5x \]

3. **Rearranging the equation**:
\[ x^2 - 5x + 1 = 0 \]

4. **Now, we need to find the value of** \( \frac{6x}{x^2 + x + 1} \).

5. **Substituting \(x^2\) from the quadratic equation**:
From \(x^2 - 5x + 1 = 0\), we can express \(x^2\) as:
\[ x^2 = 5x - 1 \]

6. **Now substitute \(x^2\) in the denominator**:
\[ x^2 + x + 1 = (5x - 1) + x + 1 = 5x - 1 + x + 1 = 6x \]

7. **Now substitute this back into the expression**:
\[ \frac{6x}{x^2 + x + 1} = \frac{6x}{6x} \]

8. **Simplifying the expression**:
\[ \frac{6x}{6x} = 1 \]

Thus, the value of \( \frac{6x}{x^2 + x + 1} \) is \(1\).