If `2x + (1)/(3x) = 5,` the value of `(5x )/( 6x ^(2) + 20 x + 1 )` is

Options

A. `1/4`
B. `1/6`
C. `1/5`
D. `1/7`

Answer

To solve the equation \( 2x + \frac{1}{3x} = 5 \) and find the value of \( \frac{5x}{6x^2 + 20x + 1} \), we will follow these steps:

### Step 1: Solve for \( x \)

We start with the equation:

\[ 2x + \frac{1}{3x} = 5 \]

To eliminate the fraction, we can multiply the entire equation by \( 3x \):

\[ 3x(2x) + 3x\left(\frac{1}{3x}\right) = 3x(5) \]

This simplifies to:

\[ 6x^2 + 1 = 15x \]

### Step 2: Rearrange the equation

Next, we rearrange the equation to set it to zero:

\[ 6x^2 - 15x + 1 = 0 \]

### Step 3: Use the quadratic formula

We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 6 \), \( b = -15 \), and \( c = 1 \):

\[ b^2 - 4ac = (-15)^2 - 4 \cdot 6 \cdot 1 = 225 - 24 = 201 \]

Now substituting into the formula:

\[ x = \frac{15 \pm \sqrt{201}}{12} \]

### Step 4: Substitute \( x \) into the expression

Now we need to find the value of:

\[ \frac{5x}{6x^2 + 20x + 1} \]

First, we calculate \( 6x^2 + 20x + 1 \):

Using our earlier equation \( 6x^2 - 15x + 1 = 0 \), we can express \( 6x^2 \) as:

\[ 6x^2 = 15x - 1 \]

Substituting this into \( 6x^2 + 20x + 1 \):

\[ 6x^2 + 20x + 1 = (15x - 1) + 20x + 1 = 35x \]

### Step 5: Calculate the final expression

Now substituting back into our expression:

\[ \frac{5x}{6x^2 + 20x + 1} = \frac{5x}{35x} \]

This simplifies to:

\[ \frac{5}{35} = \frac{1}{7} \]

### Final Answer

Thus, the value of \( \frac{5x}{6x^2 + 20x + 1} \) is:

\[ \frac{1}{7} \]