What is the mean proportional of
`3 + sqrt2 , 12 - sqrt32 `

Options

A. `4-2 sqrt2`
B. `2sqrt7`
C. `2sqrt5`
D. `6`

Answer

To find the mean proportional of the two numbers \(3 + \sqrt{2}\) and \(12 - \sqrt{32}\), we can follow these steps:

### Step 1: Identify the two numbers
Let:
- \( a = 3 + \sqrt{2} \)
- \( b = 12 - \sqrt{32} \)

### Step 2: Simplify \( b \)
First, simplify \( b \):
\[ \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \]
Thus,
\[ b = 12 - 4\sqrt{2} \]

### Step 3: Calculate the mean proportional
The mean proportional \( k \) of two numbers \( a \) and \( b \) is given by:
\[ k = \sqrt{a \cdot b} \]
Substituting the values of \( a \) and \( b \):
\[ k = \sqrt{(3 + \sqrt{2})(12 - 4\sqrt{2})} \]

### Step 4: Expand the product
Now, we will expand the product inside the square root:
\[ (3 + \sqrt{2})(12 - 4\sqrt{2}) = 3 \cdot 12 + 3 \cdot (-4\sqrt{2}) + \sqrt{2} \cdot 12 + \sqrt{2} \cdot (-4\sqrt{2}) \]
Calculating each term:
\[ = 36 - 12\sqrt{2} + 12\sqrt{2} - 4 \cdot 2 \]
\[ = 36 - 8 \]
\[ = 28 \]

### Step 5: Take the square root
Now, substitute back into the mean proportional formula:
\[ k = \sqrt{28} \]

### Step 6: Simplify the square root
We can simplify \( \sqrt{28} \):
\[ \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \]

### Final Answer
Thus, the mean proportional of \( 3 + \sqrt{2} \) and \( 12 - \sqrt{32} \) is:
\[ \boxed{2\sqrt{7}} \]

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