Solve the following :
`sqrt(x+5)+sqrt(x+12)=sqrt(2x+41)`

Answer

To solve the equation \( \sqrt{x+5} + \sqrt{x+12} = \sqrt{2x+41} \), we will follow these steps:

### Step 1: Square both sides
Start by squaring both sides of the equation to eliminate the square roots.

\[ (\sqrt{x+5} + \sqrt{x+12})^2 = (\sqrt{2x+41})^2 \]

### Step 2: Expand both sides
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand the left side:

\[ (x+5) + 2\sqrt{(x+5)(x+12)} + (x+12) = 2x + 41 \]

This simplifies to:

\[ 2x + 17 + 2\sqrt{(x+5)(x+12)} = 2x + 41 \]

### Step 3: Isolate the square root
Next, we isolate the square root term:

\[ 2\sqrt{(x+5)(x+12)} = 41 - 17 \]

This simplifies to:

\[ 2\sqrt{(x+5)(x+12)} = 24 \]

### Step 4: Divide by 2
Now, divide both sides by 2:

\[ \sqrt{(x+5)(x+12)} = 12 \]

### Step 5: Square both sides again
Square both sides again to eliminate the square root:

\[ (x+5)(x+12) = 144 \]

### Step 6: Expand and rearrange
Expand the left side:

\[ x^2 + 12x + 5x + 60 = 144 \]

This simplifies to:

\[ x^2 + 17x + 60 - 144 = 0 \]

Which further simplifies to:

\[ x^2 + 17x - 84 = 0 \]

### Step 7: Factor the quadratic equation
Now, we will factor the quadratic equation. We need two numbers that multiply to \(-84\) and add to \(17\). The numbers are \(21\) and \(-4\):

\[ (x + 21)(x - 4) = 0 \]

### Step 8: Solve for \(x\)
Set each factor to zero:

1. \(x + 21 = 0 \Rightarrow x = -21\)
2. \(x - 4 = 0 \Rightarrow x = 4\)

### Step 9: Check for extraneous solutions
We must check both solutions in the original equation to ensure they are valid:

1. For \(x = -21\):
\[ \sqrt{-21 + 5} + \sqrt{-21 + 12} = \sqrt{-42 + 41} \]
This results in complex numbers, so \(x = -21\) is not a valid solution.

2. For \(x = 4\):
\[ \sqrt{4 + 5} + \sqrt{4 + 12} = \sqrt{2(4) + 41} \]
\[ \sqrt{9} + \sqrt{16} = \sqrt{49} \]
\[ 3 + 4 = 7 \]
This is valid.

### Final Answer
The only solution to the equation is:

\[ \boxed{4} \]