`sqrt((3x)/(x+1))+sqrt((x+1)/(3x))=2," when "x ne 0and x ne -1`

Answer

To solve the equation \(\sqrt{\frac{3x}{x+1}} + \sqrt{\frac{x+1}{3x}} = 2\), we will follow these steps:

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square roots:
\[ \left(\sqrt{\frac{3x}{x+1}} + \sqrt{\frac{x+1}{3x}}\right)^2 = 2^2 \]
This simplifies to:
\[ \frac{3x}{x+1} + 2\sqrt{\frac{3x}{x+1} \cdot \frac{x+1}{3x}} + \frac{x+1}{3x} = 4 \]

### Step 2: Simplify the middle term
The middle term simplifies as follows:
\[ 2\sqrt{\frac{3x}{x+1} \cdot \frac{x+1}{3x}} = 2\sqrt{1} = 2 \]
Thus, our equation now reads:
\[ \frac{3x}{x+1} + 2 + \frac{x+1}{3x} = 4 \]

### Step 3: Rearranging the equation
Now, we can rearrange the equation:
\[ \frac{3x}{x+1} + \frac{x+1}{3x} = 4 - 2 \]
This simplifies to:
\[ \frac{3x}{x+1} + \frac{x+1}{3x} = 2 \]

### Step 4: Find a common denominator
The common denominator for the left-hand side is \(3x(x+1)\):
\[ \frac{3x \cdot 3x + (x+1)(x+1)}{3x(x+1)} = 2 \]
This leads to:
\[ \frac{9x^2 + (x^2 + 2x + 1)}{3x(x+1)} = 2 \]
Combining the terms in the numerator gives:
\[ \frac{10x^2 + 2x + 1}{3x(x+1)} = 2 \]

### Step 5: Cross-multiply
Cross-multiplying gives:
\[ 10x^2 + 2x + 1 = 6x(x+1) \]
Expanding the right side results in:
\[ 10x^2 + 2x + 1 = 6x^2 + 6x \]

### Step 6: Rearranging to form a quadratic equation
Rearranging terms leads to:
\[ 10x^2 - 6x^2 + 2x - 6x + 1 = 0 \]
This simplifies to:
\[ 4x^2 - 4x + 1 = 0 \]

### Step 7: Solve the quadratic equation
Now we can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 4\), \(b = -4\), and \(c = 1\):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]
This simplifies to:
\[ x = \frac{4 \pm \sqrt{16 - 16}}{8} = \frac{4 \pm 0}{8} = \frac{4}{8} = \frac{1}{2} \]

### Step 8: Verify the solution
We check if \(x = \frac{1}{2}\) satisfies the original equation:
\[ \sqrt{\frac{3 \cdot \frac{1}{2}}{\frac{1}{2}+1}} + \sqrt{\frac{\frac{1}{2}+1}{3 \cdot \frac{1}{2}}} = \sqrt{\frac{\frac{3}{2}}{\frac{3}{2}}} + \sqrt{\frac{\frac{3}{2}}{\frac{3}{2}}} = 1 + 1 = 2 \]
Thus, the solution \(x = \frac{1}{2}\) is valid.

### Final Answer
The solution to the equation is:
\[ x = \frac{1}{2} \]