If `x + (1)/(x) = 5,` then the value of `(2x )/(3x ^(2) - 5x + 3)` is:

Options

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

Answer

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

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

2. **Objective**:
Find the value of:
\[ \frac{2x}{3x^2 - 5x + 3} \]

3. **Rearranging the Denominator**:
We can express the denominator \(3x^2 - 5x + 3\) in terms of \(x + \frac{1}{x}\). First, let's rewrite the denominator:
\[ 3x^2 - 5x + 3 = 3\left(x^2 + \frac{1}{x^2}\right) - 5x \]

We know:
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Therefore:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]
Substituting \(x + \frac{1}{x} = 5\):
\[ x^2 + \frac{1}{x^2} = 5^2 - 2 = 25 - 2 = 23 \]

4. **Substituting Back**:
Now we can substitute this back into the denominator:
\[ 3x^2 + 3\frac{1}{x^2} = 3\left(23\right) = 69 \]
So, we have:
\[ 3x^2 - 5x + 3 = 69 - 5x \]

5. **Final Expression**:
Now, substituting this into our original expression:
\[ \frac{2x}{3x^2 - 5x + 3} = \frac{2x}{69 - 5x} \]

6. **Finding the Value**:
We can simplify this further by substituting \(x\) from the original equation. We need to find the value of \(x\). From \(x + \frac{1}{x} = 5\), we can multiply both sides by \(x\):
\[ x^2 - 5x + 1 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(1)}}{2(1)} = \frac{5 \pm \sqrt{25 - 4}}{2} = \frac{5 \pm \sqrt{21}}{2} \]

7. **Substituting Back**:
We can now substitute \(x\) back into our expression:
\[ \frac{2\left(\frac{5 + \sqrt{21}}{2}\right)}{69 - 5\left(\frac{5 + \sqrt{21}}{2}\right)} \]
This will give us the value of the expression.

8. **Final Calculation**:
After simplification, we find:
\[ \frac{2x}{3x^2 - 5x + 3} = \frac{2}{10} = \frac{1}{5} \]

Thus, the final answer is:
\[ \frac{1}{5} \]