If `(5+2sqrt6)^(x^(2)-8)+(5-2sqrt6)^(x^(2)-8)=10, x inR`
On the basis of above information, answer the following questions :
Number of solution (s) of the given equation is/ are -

Options

A. 1
B. 2
C. 4
D. infinte

Answer

To solve the equation \((5 + 2\sqrt{6})^{x^{2}-8} + (5 - 2\sqrt{6})^{x^{2}-8} = 10\), we will follow these steps:

### Step 1: Define Variables
Let \( y = x^2 - 8 \). Then, the equation can be rewritten as:
\[ (5 + 2\sqrt{6})^y + (5 - 2\sqrt{6})^y = 10 \]

### Step 2: Analyze the Terms
Notice that \( 5 + 2\sqrt{6} \) and \( 5 - 2\sqrt{6} \) are conjugates. The term \( 5 + 2\sqrt{6} \) is greater than 1, while \( 5 - 2\sqrt{6} \) is less than 1 because \( 5 - 2\sqrt{6} \approx 0.1 \).

### Step 3: Properties of the Function
The function \( f(y) = (5 + 2\sqrt{6})^y + (5 - 2\sqrt{6})^y \) is continuous and strictly increasing for \( y \in \mathbb{R} \) because the first term increases and the second term decreases as \( y \) increases.

### Step 4: Find Values of \( y \)
We need to find the values of \( y \) such that:
\[ f(y) = 10 \]
Since \( f(y) \) is continuous and strictly increasing, it will intersect the line \( y = 10 \) at most once.

### Step 5: Check Specific Values
Let's check \( y = 1 \) and \( y = -1 \):
1. For \( y = 1 \):
\[ f(1) = (5 + 2\sqrt{6})^1 + (5 - 2\sqrt{6})^1 = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10 \]
So, \( y = 1 \) is a solution.

2. For \( y = -1 \):
\[ f(-1) = (5 + 2\sqrt{6})^{-1} + (5 - 2\sqrt{6})^{-1} \]
This can be calculated as:
\[ f(-1) = \frac{1}{5 + 2\sqrt{6}} + \frac{1}{5 - 2\sqrt{6}} \]
Rationalizing gives:
\[ f(-1) = \frac{(5 - 2\sqrt{6}) + (5 + 2\sqrt{6})}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})} = \frac{10}{25 - 24} = 10 \]
So, \( y = -1 \) is also a solution.

### Step 6: Solve for \( x \)
Now we have two values for \( y \):
1. \( y = 1 \) gives:
\[ x^2 - 8 = 1 \implies x^2 = 9 \implies x = \pm 3 \]

2. \( y = -1 \) gives:
\[ x^2 - 8 = -1 \implies x^2 = 7 \implies x = \pm \sqrt{7} \]

### Step 7: Count the Solutions
Thus, we have four solutions:
- \( x = 3 \)
- \( x = -3 \)
- \( x = \sqrt{7} \)
- \( x = -\sqrt{7} \)

### Final Answer
The number of solutions of the given equation is **4**.

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