If `(5+2sqrt6)^(x^(2)-8)+(5-2sqrt6)^(x^(2)-8)=10, x inR`
On the basis of above information, answer the following questions :
If `x in (-3,5)`, then number of possible values of x, is-

Options

A. 1
B. 2
C. 3
D. 4

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 the Variables
Let \(y = x^2 - 8\). Then the equation becomes:
\[ (5 + 2\sqrt{6})^y + (5 - 2\sqrt{6})^y = 10 \]

### Step 2: Analyze the Terms
Notice that \(5 + 2\sqrt{6}\) and \(5 - 2\sqrt{6}\) are conjugates. We can denote:
\[ a = 5 + 2\sqrt{6} \quad \text{and} \quad b = 5 - 2\sqrt{6} \]
Thus, the equation can be rewritten as:
\[ a^y + b^y = 10 \]

### Step 3: Find the Relationship Between \(a\) and \(b\)
Calculate \(ab\):
\[ ab = (5 + 2\sqrt{6})(5 - 2\sqrt{6}) = 25 - (2\sqrt{6})^2 = 25 - 24 = 1 \]
This means \(b = \frac{1}{a}\).

### Step 4: Substitute \(b\) in the Equation
Substituting \(b\) gives:
\[ a^y + \left(\frac{1}{a}\right)^y = 10 \]
This simplifies to:
\[ a^y + \frac{1}{a^y} = 10 \]

### Step 5: Let \(z = a^y\)
Then the equation becomes:
\[ z + \frac{1}{z} = 10 \]
Multiplying through by \(z\) gives:
\[ z^2 - 10z + 1 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula:
\[ z = \frac{10 \pm \sqrt{10^2 - 4 \cdot 1}}{2 \cdot 1} = \frac{10 \pm \sqrt{96}}{2} = \frac{10 \pm 4\sqrt{6}}{2} = 5 \pm 2\sqrt{6} \]

### Step 7: Find \(y\)
Since \(z = a^y\), we have two cases:
1. \(a^y = 5 + 2\sqrt{6}\)
2. \(a^y = 5 - 2\sqrt{6}\)

For the first case:
\[ y = 1 \implies x^2 - 8 = 1 \implies x^2 = 9 \implies x = 3 \text{ or } x = -3 \]

For the second case:
\[ y = -1 \implies x^2 - 8 = -1 \implies x^2 = 7 \implies x = \sqrt{7} \text{ or } x = -\sqrt{7} \]

### Step 8: Identify Valid Solutions in the Interval
Now we need to check which of these solutions lie in the interval \((-3, 5)\):
- From \(y = 1\): \(x = 3\) (valid), \(x = -3\) (not valid).
- From \(y = -1\): \(x = \sqrt{7} \approx 2.645\) (valid), \(x = -\sqrt{7} \approx -2.645\) (valid).

### Conclusion
The valid solutions in the interval \((-3, 5)\) are:
- \(x = 3\)
- \(x = \sqrt{7}\)
- \(x = -\sqrt{7}\)

Thus, the total number of possible values of \(x\) is **3**.