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

Options

A. `5/ 2`
B. `25/3`
C. `4`
D. `3 `

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Cross-multiply
We can cross-multiply to eliminate the fraction:

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

### Step 2: Expand the right side
Distributing the 3 on the right side gives:

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

### Step 3: Move all terms involving square roots to one side
Rearranging the equation, we have:

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

### Step 4: Combine like terms
Now, let's move all terms involving \(\sqrt{5 + x}\) to one side and \(\sqrt{5 - x}\) to the other side:

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

This simplifies to:

\[ 4\sqrt{5 - x} = 2\sqrt{5 + x}. \]

### Step 5: Simplify the equation
Dividing both sides by 2 gives us:

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

### Step 6: Square both sides
Squaring both sides results in:

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

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

\[ 20 - 4x = 5 + x. \]

Now, rearranging the equation to isolate \(x\):

\[ 20 - 5 = x + 4x, \]

which simplifies to:

\[ 15 = 5x. \]

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

\[ x = 3. \]

Thus, the value of \(x\) is:

\[ \boxed{3}. \]

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