If `x=1+sqrt(2)+sqrt(3)`, then find the value of `x^(2)-2x+4`.

Options

A. `2(7+sqrt(6))`
B. `2(4+sqrt(6))`
C. `2(3+sqrt(6))`
D. `(4+sqrt(6))`

Answer

To find the value of \( x^2 - 2x + 4 \) given \( x = 1 + \sqrt{2} + \sqrt{3} \), we can follow these steps:

### Step 1: Calculate \( x^2 \)

First, we need to calculate \( x^2 \):
\[ x = 1 + \sqrt{2} + \sqrt{3} \]
Now, squaring \( x \):
\[ x^2 = (1 + \sqrt{2} + \sqrt{3})^2 \]
Using the formula \( (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \), we have:
\[ x^2 = 1^2 + (\sqrt{2})^2 + (\sqrt{3})^2 + 2(1 \cdot \sqrt{2} + 1 \cdot \sqrt{3} + \sqrt{2} \cdot \sqrt{3}) \]
Calculating each term:
\[ 1^2 = 1, \quad (\sqrt{2})^2 = 2, \quad (\sqrt{3})^2 = 3 \]
Thus,
\[ x^2 = 1 + 2 + 3 + 2(\sqrt{2} + \sqrt{3} + \sqrt{6}) \]
This simplifies to:
\[ x^2 = 6 + 2(\sqrt{2} + \sqrt{3} + \sqrt{6}) \]

### Step 2: Calculate \( 2x \)

Next, we calculate \( 2x \):
\[ 2x = 2(1 + \sqrt{2} + \sqrt{3}) = 2 + 2\sqrt{2} + 2\sqrt{3} \]

### Step 3: Substitute into the expression \( x^2 - 2x + 4 \)

Now, we substitute \( x^2 \) and \( 2x \) into the expression:
\[ x^2 - 2x + 4 = (6 + 2(\sqrt{2} + \sqrt{3} + \sqrt{6})) - (2 + 2\sqrt{2} + 2\sqrt{3}) + 4 \]
Combining the constants:
\[ = 6 - 2 + 4 + 2(\sqrt{2} + \sqrt{3} + \sqrt{6}) - 2\sqrt{2} - 2\sqrt{3} \]
This simplifies to:
\[ = 8 + 2(\sqrt{6}) \quad \text{(since \(2\sqrt{2}\) and \(2\sqrt{3}\) cancel out)} \]

### Step 4: Factor out the common term

Finally, we can factor out the common term:
\[ = 2(4 + \sqrt{6}) \]

Thus, the value of \( x^2 - 2x + 4 \) is:
\[ \boxed{2(4 + \sqrt{6})} \]