If `(7-4sqrt(3))^(x^(2)-4x+3)+(7+4sqrt(3))^(x^(2)-4x+3)=14`, then the value of `x` is given by

Options

A. `2`
B. `2-sqrt(2)`
C. `2+sqrt(2)`
D. none of these

Answer

To solve the equation \( (7 - 4\sqrt{3})^{x^2 - 4x + 3} + (7 + 4\sqrt{3})^{x^2 - 4x + 3} = 14 \), we will follow these steps:

### Step 1: Define a substitution
Let \( t = x^2 - 4x + 3 \). Then, we can rewrite the equation as:
\[ (7 - 4\sqrt{3})^t + (7 + 4\sqrt{3})^t = 14 \]

### Step 2: Simplify the bases
We can calculate \( (7 - 4\sqrt{3})(7 + 4\sqrt{3}) \):
\[ (7 - 4\sqrt{3})(7 + 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]
This means that:
\[ (7 + 4\sqrt{3}) = \frac{1}{(7 - 4\sqrt{3})} \]

### Step 3: Rewrite the equation
Using the relationship from Step 2, we can rewrite the equation:
\[ (7 - 4\sqrt{3})^t + \left(\frac{1}{(7 - 4\sqrt{3})}\right)^t = 14 \]
Let \( a = (7 - 4\sqrt{3})^t \). Then the equation becomes:
\[ a + \frac{1}{a} = 14 \]

### Step 4: Multiply through by \( a \)
Multiply both sides by \( a \) to eliminate the fraction:
\[ a^2 + 1 = 14a \]
Rearranging gives us:
\[ a^2 - 14a + 1 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{14 \pm \sqrt{196 - 4}}{2} = \frac{14 \pm \sqrt{192}}{2} \]
\[ = \frac{14 \pm 8\sqrt{3}}{2} = 7 \pm 4\sqrt{3} \]

### Step 6: Determine the values of \( t \)
Since \( a = (7 - 4\sqrt{3})^t \), we have two cases:
1. \( (7 - 4\sqrt{3})^t = 7 - 4\sqrt{3} \)
2. \( (7 - 4\sqrt{3})^t = 7 + 4\sqrt{3} \)

#### Case 1:
For \( (7 - 4\sqrt{3})^t = 7 - 4\sqrt{3} \):
This implies \( t = 1 \).

#### Case 2:
For \( (7 - 4\sqrt{3})^t = 7 + 4\sqrt{3} \):
This implies \( t = -1 \).

### Step 7: Solve for \( x \)
Now we need to solve for \( x \) using the values of \( t \):
1. If \( t = 1 \):
\[ x^2 - 4x + 3 = 1 \implies x^2 - 4x + 2 = 0 \]
Using the quadratic formula:
\[ x = \frac{4 \pm \sqrt{16 - 8}}{2} = \frac{4 \pm 2\sqrt{2}}{2} = 2 \pm \sqrt{2} \]

2. If \( t = -1 \):
\[ x^2 - 4x + 3 = -1 \implies x^2 - 4x + 4 = 0 \implies (x - 2)^2 = 0 \implies x = 2 \]

### Final Solutions
The possible values of \( x \) are:
\[ x = 2 + \sqrt{2}, \quad x = 2 - \sqrt{2}, \quad x = 2 \]