Using properties of proportion, solve for x. Given that x is positive :
`(2x+sqrt(4x^(2)-1))/(2x-sqrt(4x^(2)-1))=4`

Answer

To solve the equation \((2x + \sqrt{4x^2 - 1})/(2x - \sqrt{4x^2 - 1}) = 4\) using properties of proportion, we will follow these steps:

### Step 1: Set up the equation
We start with the given equation:
\[ \frac{2x + \sqrt{4x^2 - 1}}{2x - \sqrt{4x^2 - 1}} = 4 \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ 2x + \sqrt{4x^2 - 1} = 4(2x - \sqrt{4x^2 - 1}) \]
Expanding the right side:
\[ 2x + \sqrt{4x^2 - 1} = 8x - 4\sqrt{4x^2 - 1} \]

### Step 3: Rearrange the equation
Now, we will rearrange the equation to isolate the square root terms:
\[ \sqrt{4x^2 - 1} + 4\sqrt{4x^2 - 1} = 8x - 2x \]
This simplifies to:
\[ 5\sqrt{4x^2 - 1} = 6x \]

### Step 4: Isolate the square root
Now, divide both sides by 5:
\[ \sqrt{4x^2 - 1} = \frac{6x}{5} \]

### Step 5: Square both sides
Next, we square both sides to eliminate the square root:
\[ 4x^2 - 1 = \left(\frac{6x}{5}\right)^2 \]
This gives us:
\[ 4x^2 - 1 = \frac{36x^2}{25} \]

### Step 6: Clear the fraction
To eliminate the fraction, multiply through by 25:
\[ 25(4x^2 - 1) = 36x^2 \]
Expanding gives:
\[ 100x^2 - 25 = 36x^2 \]

### Step 7: Rearrange the equation
Now, we will rearrange the equation:
\[ 100x^2 - 36x^2 = 25 \]
This simplifies to:
\[ 64x^2 = 25 \]

### Step 8: Solve for x
Now, divide both sides by 64:
\[ x^2 = \frac{25}{64} \]
Taking the square root gives:
\[ x = \frac{5}{8} \quad \text{(since x is positive)} \]

### Final Answer
Thus, the solution for \(x\) is:
\[ \boxed{\frac{5}{8}} \]