If `(x^(3) +3x)/(3x^(2)+1)=(189)/(61)` then value of x is

Options

A. 9
B. 11
C. 7
D. 13

Answer

To solve the equation

\[ \frac{x^3 + 3x}{3x^2 + 1} = \frac{189}{61} \]

we will follow these steps:

### Step 1: Cross-Multiply
Cross-multiply to eliminate the fractions.

\[ (x^3 + 3x) \cdot 61 = (3x^2 + 1) \cdot 189 \]

### Step 2: Expand Both Sides
Now, expand both sides of the equation.

\[ 61x^3 + 183x = 567x^2 + 189 \]

### Step 3: Rearrange the Equation
Rearranging the equation to bring all terms to one side gives:

\[ 61x^3 - 567x^2 + 183x - 189 = 0 \]

### Step 4: Factor or Use the Rational Root Theorem
We can try to find rational roots using the Rational Root Theorem or by inspection. Testing \(x = 3\):

\[ 61(3)^3 - 567(3)^2 + 183(3) - 189 = 61(27) - 567(9) + 549 - 189 \]

Calculating each term:

\[ 61 \cdot 27 = 1647 \]
\[ 567 \cdot 9 = 5103 \]
\[ 183 \cdot 3 = 549 \]

So we have:

\[ 1647 - 5103 + 549 - 189 = 0 \]

This confirms that \(x = 3\) is a root.

### Step 5: Conclusion
The value of \(x\) is:

\[ \boxed{3} \]

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