If `2x + (1)/( 3x ) = 6,` then `3x + (1)/(2x)` is equal to

Options

A. 4
B. 8
C. 9
D. 12

Answer

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

### Step 1: Solve for \( x \) from the first equation
We start with the equation:
\[ 2x + \frac{1}{3x} = 6 \]
To eliminate the fraction, we can multiply the entire equation by \( 3x \) (the LCM of the denominators):
\[ 3x(2x) + 3x\left(\frac{1}{3x}\right) = 6 \cdot 3x \]
This simplifies to:
\[ 6x^2 + 1 = 18x \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to form a standard quadratic equation:
\[ 6x^2 - 18x + 1 = 0 \]

### Step 3: Use the quadratic formula
To solve for \( x \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 6 \), \( b = -18 \), and \( c = 1 \). Plugging in these values, we get:
\[ x = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 6 \cdot 1}}{2 \cdot 6} \]
Calculating the discriminant:
\[ (-18)^2 = 324 \]
\[ 4 \cdot 6 \cdot 1 = 24 \]
Thus:
\[ b^2 - 4ac = 324 - 24 = 300 \]
Now substituting back into the formula:
\[ x = \frac{18 \pm \sqrt{300}}{12} \]
Simplifying \( \sqrt{300} = 10\sqrt{3} \):
\[ x = \frac{18 \pm 10\sqrt{3}}{12} = \frac{3 \pm \frac{5\sqrt{3}}{3}}{2} \]

### Step 4: Find \( 3x + \frac{1}{2x} \)
Now we need to find \( 3x + \frac{1}{2x} \). We can express this in terms of \( 6x^2 + 1 \):
\[ 3x + \frac{1}{2x} = \frac{6x^2 + 1}{2x} \]
From our previous step, we know \( 6x^2 + 1 = 18x \):
\[ 3x + \frac{1}{2x} = \frac{18x}{2x} = 9 \]

### Final Answer
Thus, the value of \( 3x + \frac{1}{2x} \) is:
\[ \boxed{9} \]