If `(sqrt(x+5)+sqrt(x))/(sqrt(x+5)-sqrt(x))=5` then find the value of 'x'.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation

\[ \frac{\sqrt{x+5} + \sqrt{x}}{\sqrt{x+5} - \sqrt{x}} = 5, \]

we can use the concept of component and dividend theorem. Let's break down the solution step by step.

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fraction:

\[ \sqrt{x+5} + \sqrt{x} = 5(\sqrt{x+5} - \sqrt{x}). \]

### Step 2: Distribute on the Right Side
Distributing the 5 on the right side gives us:

\[ \sqrt{x+5} + \sqrt{x} = 5\sqrt{x+5} - 5\sqrt{x}. \]

### Step 3: Rearrange the Equation
Now, let's move all terms involving square roots to one side:

\[ \sqrt{x+5} + \sqrt{x} - 5\sqrt{x+5} + 5\sqrt{x} = 0. \]

This simplifies to:

\[ -4\sqrt{x+5} + 6\sqrt{x} = 0. \]

### Step 4: Isolate One Square Root
Rearranging gives:

\[ 4\sqrt{x+5} = 6\sqrt{x}. \]

### Step 5: Divide Both Sides by 2
Dividing both sides by 2:

\[ 2\sqrt{x+5} = 3\sqrt{x}. \]

### Step 6: Square Both Sides
Now, square both sides to eliminate the square roots:

\[ (2\sqrt{x+5})^2 = (3\sqrt{x})^2. \]

This results in:

\[ 4(x+5) = 9x. \]

### Step 7: Expand and Rearrange
Expanding the left side gives:

\[ 4x + 20 = 9x. \]

Rearranging this leads to:

\[ 20 = 9x - 4x, \]
\[ 20 = 5x. \]

### Step 8: Solve for x
Dividing both sides by 5 gives:

\[ x = 4. \]

### Step 9: Verify the Solution
To ensure that our solution is correct, we can substitute \(x = 4\) back into the original equation:

\[ \sqrt{4+5} + \sqrt{4} = \sqrt{9} + 2 = 3 + 2 = 5, \]
\[ \sqrt{4+5} - \sqrt{4} = \sqrt{9} - 2 = 3 - 2 = 1. \]

Thus,

\[ \frac{5}{1} = 5, \]

which confirms our solution.

### Final Answer
The value of \(x\) is

\[ \boxed{4}. \]