If `sqrt(3x - 7) + sqrt(3x + 7) = 4 + sqrt(2)` then value of `x + 1/x` is

Options

A. A) `82/9`
B. B) `10/3`
C. C) `5/2`
D. D) `4/3`

Answer

To solve the equation \( \sqrt{3x - 7} + \sqrt{3x + 7} = 4 + \sqrt{2} \), we will follow these steps:

### Step 1: Isolate one of the square roots
We can rewrite the equation as:
\[ \sqrt{3x + 7} = (4 + \sqrt{2}) - \sqrt{3x - 7} \]

### Step 2: Square both sides
Now, we will square both sides to eliminate the square roots:
\[ 3x + 7 = \left( (4 + \sqrt{2}) - \sqrt{3x - 7} \right)^2 \]

### Step 3: Expand the right-hand side
Expanding the right-hand side:
\[ 3x + 7 = (4 + \sqrt{2})^2 - 2(4 + \sqrt{2})\sqrt{3x - 7} + (3x - 7) \]
Calculating \( (4 + \sqrt{2})^2 \):
\[ (4 + \sqrt{2})^2 = 16 + 8\sqrt{2} + 2 = 18 + 8\sqrt{2} \]
So we have:
\[ 3x + 7 = 18 + 8\sqrt{2} - 2(4 + \sqrt{2})\sqrt{3x - 7} + 3x - 7 \]
This simplifies to:
\[ 3x + 7 = 3x + 11 + 8\sqrt{2} - 2(4 + \sqrt{2})\sqrt{3x - 7} \]

### Step 4: Simplify the equation
Subtract \( 3x \) from both sides:
\[ 7 = 11 + 8\sqrt{2} - 2(4 + \sqrt{2})\sqrt{3x - 7} \]
Rearranging gives:
\[ 2(4 + \sqrt{2})\sqrt{3x - 7} = 11 + 8\sqrt{2} - 7 \]
\[ 2(4 + \sqrt{2})\sqrt{3x - 7} = 4 + 8\sqrt{2} \]

### Step 5: Divide both sides by \( 2(4 + \sqrt{2}) \)
\[ \sqrt{3x - 7} = \frac{4 + 8\sqrt{2}}{2(4 + \sqrt{2})} \]
This simplifies to:
\[ \sqrt{3x - 7} = \frac{2 + 4\sqrt{2}}{4 + \sqrt{2}} \]

### Step 6: Square both sides again
Squaring both sides again:
\[ 3x - 7 = \left(\frac{2 + 4\sqrt{2}}{4 + \sqrt{2}}\right)^2 \]

### Step 7: Solve for \( x \)
After calculating the right-hand side and simplifying, we find:
\[ 3x = 9 \implies x = 3 \]

### Step 8: Find \( x + \frac{1}{x} \)
Now, we calculate:
\[ x + \frac{1}{x} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \]

### Final Answer
Thus, the value of \( x + \frac{1}{x} \) is:
\[ \frac{10}{3} \]

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