Find number of solutions of the equation `sqrt((x+8)+2sqrt(x+7))+sqrt((x+1)-sqrt(x+7))=4`

Explanation

To find the number of solutions of the equation

\[ \sqrt{(x+8) + 2\sqrt{x+7}} + \sqrt{(x+1) - \sqrt{x+7}} = 4, \]

we will follow these steps:

### Step 1: Substitute Variables
Let \( t = \sqrt{x + 7} \). Then, we can express \( x \) in terms of \( t \):
\[ x + 7 = t^2 \implies x = t^2 - 7. \]
Now we can express \( x + 8 \) and \( x + 1 \) as follows:
\[ x + 8 = t^2 - 7 + 8 = t^2 + 1, \]
\[ x + 1 = t^2 - 7 + 1 = t^2 - 6. \]

### Step 2: Rewrite the Original Equation
Substituting these values into the original equation gives:
\[ \sqrt{(t^2 + 1) + 2t} + \sqrt{(t^2 - 6) - t} = 4. \]
This simplifies to:
\[ \sqrt{t^2 + 2t + 1} + \sqrt{t^2 - 6 - t} = 4. \]

### Step 3: Simplify the Square Roots
Notice that:
\[ \sqrt{t^2 + 2t + 1} = \sqrt{(t + 1)^2} = t + 1. \]
Thus, the equation becomes:
\[ t + 1 + \sqrt{t^2 - t - 6} = 4. \]

### Step 4: Isolate the Square Root
Rearranging gives:
\[ \sqrt{t^2 - t - 6} = 4 - (t + 1) = 3 - t. \]

### Step 5: Square Both Sides
Now, we square both sides to eliminate the square root:
\[ t^2 - t - 6 = (3 - t)^2. \]
Expanding the right side:
\[ t^2 - t - 6 = 9 - 6t + t^2. \]

### Step 6: Simplify the Equation
Subtract \( t^2 \) from both sides:
\[ -t - 6 = 9 - 6t. \]
Rearranging gives:
\[ 6t - t = 9 + 6 \implies 5t = 15 \implies t = 3. \]

### Step 7: Back Substitute for \( x \)
Recall that \( t = \sqrt{x + 7} \). Thus:
\[ \sqrt{x + 7} = 3. \]
Squaring both sides results in:
\[ x + 7 = 9 \implies x = 2. \]

### Step 8: Verify the Solution
We need to check if \( x = 2 \) satisfies the original equation:
\[ \sqrt{(2 + 8) + 2\sqrt{2 + 7}} + \sqrt{(2 + 1) - \sqrt{2 + 7}} = \sqrt{10 + 6} + \sqrt{3 - 3} = \sqrt{16} + 0 = 4. \]
Since both sides are equal, \( x = 2 \) is indeed a solution.

### Conclusion
Thus, the number of solutions to the equation is **1**.

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