The number of real solutions the equation `sqrt(x+14-8sqrt(x-2))+sqrt(x+23-10sqrt(x-2))=3` are

Options

A. 2
B. 4
C. 8
D. infinite

Answer

To find the number of real solutions for the equation

\[ \sqrt{x + 14 - 8\sqrt{x - 2}} + \sqrt{x + 23 - 10\sqrt{x - 2}} = 3, \]

we can follow these steps:

### Step 1: Rewrite the equation
We can rewrite \(x + 14\) and \(x + 23\) in terms of \(\sqrt{x - 2}\).

\[ x + 14 = (x - 2) + 16 \quad \text{and} \quad x + 23 = (x - 2) + 25. \]

Thus, we can rewrite the equation as:

\[ \sqrt{(x - 2) + 16 - 8\sqrt{x - 2}} + \sqrt{(x - 2) + 25 - 10\sqrt{x - 2}} = 3. \]

### Step 2: Substitute \(y = \sqrt{x - 2}\)
Let \(y = \sqrt{x - 2}\). Then \(x = y^2 + 2\). Substitute \(y\) into the equation:

\[ \sqrt{y^2 + 2 + 14 - 8y} + \sqrt{y^2 + 2 + 23 - 10y} = 3. \]

This simplifies to:

\[ \sqrt{y^2 - 8y + 16} + \sqrt{y^2 - 10y + 25} = 3. \]

### Step 3: Simplify the square roots
The square roots can be simplified:

\[ \sqrt{(y - 4)^2} + \sqrt{(y - 5)^2} = 3. \]

This leads to:

\[ |y - 4| + |y - 5| = 3. \]

### Step 4: Analyze cases based on the absolute values
We will consider three cases based on the values of \(y\).

#### Case 1: \(y < 4\)
In this case, both expressions inside the absolute values are negative:

\[ -(y - 4) - (y - 5) = 3 \implies -2y + 9 = 3 \implies -2y = -6 \implies y = 3. \]

Since \(y = 3 < 4\), this solution is valid.

#### Case 2: \(4 \leq y < 5\)
Here, \(y - 4 \geq 0\) and \(y - 5 < 0\):

\[ (y - 4) - (y - 5) = 3 \implies 1 = 3, \]

which is not possible. Thus, there are no solutions in this case.

#### Case 3: \(y \geq 5\)
In this case, both expressions are positive:

\[ (y - 4) + (y - 5) = 3 \implies 2y - 9 = 3 \implies 2y = 12 \implies y = 6. \]

Since \(y = 6 \geq 5\), this solution is valid.

### Step 5: Convert back to \(x\)
Now we convert the valid \(y\) values back to \(x\):

1. From \(y = 3\):
\[ \sqrt{x - 2} = 3 \implies x - 2 = 9 \implies x = 11. \]

2. From \(y = 6\):
\[ \sqrt{x - 2} = 6 \implies x - 2 = 36 \implies x = 38. \]

### Conclusion
Thus, we have found two real solutions for the equation:

- \(x = 11\)
- \(x = 38\)

The number of real solutions is **2**.