The number of real roots of `(7+4sqrt(3))^(|x|- 8)+(7-4sqrt(3))^(|x|-8)=14` is

Options

A. 0
B. 4
C. 1
D. None of these

Answer

To solve the equation \((7 + 4\sqrt{3})^{|x| - 8} + (7 - 4\sqrt{3})^{|x| - 8} = 14\), we will follow these steps:

### Step 1: Simplify the Equation
First, we note that:
\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 49 - 48 = 1 \]
This implies:
\[ 7 - 4\sqrt{3} = \frac{1}{7 + 4\sqrt{3}} \]
Thus, we can rewrite the equation as:
\[ (7 + 4\sqrt{3})^{|x| - 8} + \left(\frac{1}{7 + 4\sqrt{3}}\right)^{|x| - 8} = 14 \]
This simplifies to:
\[ (7 + 4\sqrt{3})^{|x| - 8} + (7 + 4\sqrt{3})^{-|x| + 8} = 14 \]

### Step 2: Let \( t = (7 + 4\sqrt{3})^{|x| - 8} \)
Now, we can substitute \( t \):
\[ t + \frac{1}{t} = 14 \]
Multiplying through by \( t \) gives:
\[ t^2 - 14t + 1 = 0 \]

### Step 3: Solve the Quadratic Equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = -14 \), and \( c = 1 \).
\[ t = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
Calculating the discriminant:
\[ t = \frac{14 \pm \sqrt{196 - 4}}{2} = \frac{14 \pm \sqrt{192}}{2} = \frac{14 \pm 8\sqrt{3}}{2} = 7 \pm 4\sqrt{3} \]

### Step 4: Find Values of \( |x| - 8 \)
Now we have two values for \( t \):
1. \( t = 7 + 4\sqrt{3} \)
2. \( t = 7 - 4\sqrt{3} \)

For each \( t \):
1. **For \( t = 7 + 4\sqrt{3} \)**:
\[ (7 + 4\sqrt{3})^{|x| - 8} = 7 + 4\sqrt{3} \]
This implies:
\[ |x| - 8 = 1 \quad \Rightarrow \quad |x| = 9 \]
Thus, \( x = 9 \) or \( x = -9 \).

2. **For \( t = 7 - 4\sqrt{3} \)**:
\[ (7 + 4\sqrt{3})^{|x| - 8} = 7 - 4\sqrt{3} \]
Since \( 7 - 4\sqrt{3} = (7 + 4\sqrt{3})^{-1} \), we have:
\[ |x| - 8 = -1 \quad \Rightarrow \quad |x| = 7 \]
Thus, \( x = 7 \) or \( x = -7 \).

### Step 5: Count the Real Roots
From both cases, we have the following values for \( x \):
- From \( |x| = 9 \): \( x = 9, -9 \)
- From \( |x| = 7 \): \( x = 7, -7 \)

Thus, the total number of real roots is:
\[ \text{Total roots} = 4 \quad (9, -9, 7, -7) \]

### Final Answer
The number of real roots of the equation is **4**.