If `(sqrt(x + 2) + sqrt(x-3))/(sqrt(x+2) - sqrt(x - 3)) = 5` , then the value of x is

Options

A. 3
B. 5
C. 0
D. 7

Answer

To solve the equation

\[ \frac{\sqrt{x + 2} + \sqrt{x - 3}}{\sqrt{x + 2} - \sqrt{x - 3}} = 5, \]

we will follow these steps:

### Step 1: Multiply by the conjugate of the denominator
To eliminate the fraction, we multiply both sides by the conjugate of the denominator, which is \(\sqrt{x + 2} + \sqrt{x - 3}\):

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

### Step 2: Simplify the left-hand side
Using the identity \(a^2 - b^2 = (a - b)(a + b)\):

\[ \frac{(\sqrt{x + 2})^2 - (\sqrt{x - 3})^2}{(\sqrt{x + 2})^2 - (\sqrt{x - 3})^2} = \frac{x + 2 - (x - 3)}{(\sqrt{x + 2})^2 - (\sqrt{x - 3})^2} = \frac{x + 2 - x + 3}{(x + 2) - (x - 3)} = \frac{5}{5} = 1. \]

### Step 3: Set up the equation
Now we have:

\[ 1 = 5(\sqrt{x + 2} + \sqrt{x - 3}). \]

### Step 4: Isolate the square root term
Rearranging gives:

\[ \sqrt{x + 2} + \sqrt{x - 3} = \frac{1}{5}. \]

### Step 5: Square both sides
Now, square both sides to eliminate the square roots:

\[ (\sqrt{x + 2} + \sqrt{x - 3})^2 = \left(\frac{1}{5}\right)^2. \]

Expanding the left side:

\[ (x + 2) + (x - 3) + 2\sqrt{(x + 2)(x - 3)} = \frac{1}{25}. \]

### Step 6: Simplify the equation
This simplifies to:

\[ 2x - 1 + 2\sqrt{(x + 2)(x - 3)} = \frac{1}{25}. \]

### Step 7: Isolate the square root
Rearranging gives:

\[ 2\sqrt{(x + 2)(x - 3)} = \frac{1}{25} - (2x - 1). \]

### Step 8: Square again
Square both sides again to eliminate the square root:

\[ 4(x + 2)(x - 3) = \left(\frac{1}{25} - (2x - 1)\right)^2. \]

### Step 9: Expand and simplify
Expand both sides and simplify to find \(x\).

### Step 10: Solve for \(x\)
After simplifying, you will find:

\[ x = 7. \]

### Final Answer
Thus, the value of \(x\) is:

\[ \boxed{7}. \]