If one root of `sqrt(a-x) + sqrt(b+x) = sqrt(a) + sqrt(b)` is 2012, then a possible value of (a, b) is:

Answer

To solve the equation \( \sqrt{a-x} + \sqrt{b+x} = \sqrt{a} + \sqrt{b} \) given that one root is \( x = 2012 \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \sqrt{a-x} + \sqrt{b+x} = \sqrt{a} + \sqrt{b} \]
Squaring both sides gives:
\[ (\sqrt{a-x} + \sqrt{b+x})^2 = (\sqrt{a} + \sqrt{b})^2 \]

### Step 2: Expand both sides
Expanding the left side:
\[ (a-x) + (b+x) + 2\sqrt{(a-x)(b+x)} = a + b + 2\sqrt{ab} \]
This simplifies to:
\[ a + b + 2\sqrt{(a-x)(b+x)} = a + b + 2\sqrt{ab} \]

### Step 3: Cancel common terms
Since \( a + b \) appears on both sides, we can cancel it:
\[ 2\sqrt{(a-x)(b+x)} = 2\sqrt{ab} \]
Dividing both sides by 2 gives:
\[ \sqrt{(a-x)(b+x)} = \sqrt{ab} \]

### Step 4: Square both sides again
Squaring both sides again results in:
\[ (a-x)(b+x) = ab \]

### Step 5: Expand and rearrange
Expanding the left side:
\[ ab + ax - bx - x^2 = ab \]
Cancelling \( ab \) from both sides gives:
\[ ax - bx - x^2 = 0 \]

### Step 6: Factor the equation
Factoring out \( x \):
\[ x(a - b) - x^2 = 0 \]
This can be rearranged as:
\[ x(x - (a - b)) = 0 \]

### Step 7: Identify the roots
From this equation, we have two roots:
1. \( x = 0 \)
2. \( x = a - b \)

Given that one root is \( x = 2012 \), we set:
\[ a - b = 2012 \]

### Step 8: Express \( a \) in terms of \( b \)
From the equation \( a - b = 2012 \), we can express \( a \) as:
\[ a = b + 2012 \]

### Step 9: Choose a value for \( b \)
To find a possible pair \( (a, b) \), we can choose a value for \( b \). Let's take \( b = 2012 \):
\[ a = 2012 + 2012 = 4024 \]

### Final Answer
Thus, one possible value of \( (a, b) \) is:
\[ (a, b) = (4024, 2012) \]