The number of real solution of `sqrt(x + 8) + sqrt( x - 1) = 9` is _____

Answer

To solve the equation \( \sqrt{x + 8} + \sqrt{x - 1} = 9 \) and find the number of real solutions, we can follow these steps:

### Step 1: Isolate one of the square root terms
We can rearrange the equation to isolate one of the square root terms:
\[ \sqrt{x + 8} = 9 - \sqrt{x - 1} \]

### Step 2: Square both sides
Next, we square both sides to eliminate the square root:
\[ (\sqrt{x + 8})^2 = (9 - \sqrt{x - 1})^2 \]
This simplifies to:
\[ x + 8 = 81 - 18\sqrt{x - 1} + (x - 1) \]

### Step 3: Simplify the equation
Now, we simplify the equation:
\[ x + 8 = 81 - 1 + x - 18\sqrt{x - 1} \]
\[ x + 8 = 80 + x - 18\sqrt{x - 1} \]

### Step 4: Eliminate \(x\) from both sides
Subtract \(x\) from both sides:
\[ 8 = 80 - 18\sqrt{x - 1} \]

### Step 5: Rearrange to isolate the square root
Now, isolate the square root:
\[ 18\sqrt{x - 1} = 80 - 8 \]
\[ 18\sqrt{x - 1} = 72 \]

### Step 6: Divide by 18
Divide both sides by 18:
\[ \sqrt{x - 1} = 4 \]

### Step 7: Square both sides again
Square both sides again to solve for \(x\):
\[ x - 1 = 16 \]
\[ x = 17 \]

### Step 8: Verify the solution
We need to check if \(x = 17\) satisfies the original equation:
\[ \sqrt{17 + 8} + \sqrt{17 - 1} = \sqrt{25} + \sqrt{16} = 5 + 4 = 9 \]
Since this holds true, \(x = 17\) is indeed a solution.

### Conclusion
Thus, the number of real solutions to the equation \( \sqrt{x + 8} + \sqrt{x - 1} = 9 \) is:
\[ \text{Number of real solutions} = 1 \]

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