Solve : `sqrt(9+2x)-sqrt(2x)=(5)/(sqrt(9+2x))` .

Answer

To solve the equation \( \sqrt{9 + 2x} - \sqrt{2x} = \frac{5}{\sqrt{9 + 2x}} \), we will follow these steps:

### Step 1: Cross-multiply
First, we will cross-multiply to eliminate the fraction. This gives us:
\[ \left( \sqrt{9 + 2x} - \sqrt{2x} \right) \cdot \sqrt{9 + 2x} = 5 \]

### Step 2: Expand the left-hand side
Expanding the left-hand side, we have:
\[ (\sqrt{9 + 2x})^2 - \sqrt{2x} \cdot \sqrt{9 + 2x} = 5 \]
This simplifies to:
\[ 9 + 2x - \sqrt{2x(9 + 2x)} = 5 \]

### Step 3: Rearrange the equation
Now, we will rearrange the equation to isolate the square root:
\[ 9 + 2x - 5 = \sqrt{2x(9 + 2x)} \]
This simplifies to:
\[ 4 + 2x = \sqrt{2x(9 + 2x)} \]

### Step 4: Square both sides
Next, we square both sides to eliminate the square root:
\[ (4 + 2x)^2 = 2x(9 + 2x) \]

### Step 5: Expand both sides
Expanding both sides gives us:
\[ 16 + 16x + 4x^2 = 18x + 4x^2 \]

### Step 6: Simplify the equation
Now, we can simplify the equation by subtracting \(4x^2\) from both sides:
\[ 16 + 16x = 18x \]
Rearranging gives:
\[ 16 = 18x - 16x \]
This simplifies to:
\[ 16 = 2x \]

### Step 7: Solve for \(x\)
Dividing both sides by 2, we find:
\[ x = 8 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{8} \]