Question

\(\text{Let } S = \{x \in \mathbb{R} : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10\}\).Then the number of elements in \( S \) is:

• A. 4
• B. 0
• C. 2
• D. 1

Answer 1

The Correct Option is C

Given the equation:

\[ \left( \sqrt{3} + \sqrt{2} \right)^x + \left( \sqrt{3} - \sqrt{2} \right)^x = 10 \]

Let \( t = \left( \sqrt{3} + \sqrt{2} \right)^x \).

Then, \( \left( \sqrt{3} - \sqrt{2} \right)^x = \frac{1}{t} \).

Substituting these into the original equation yields:

\[ t + \frac{1}{t} = 10 \]

Multiplying by \( t \):

\[ t^2 + 1 = 10t \]

Rearranging into a quadratic equation:

\[ t^2 - 10t + 1 = 0 \]

Solving for \( t \) using the quadratic formula:

\[ t = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(1)}}{2(1)} = \frac{10 \pm \sqrt{100 - 4}}{2} = \frac{10 \pm \sqrt{96}}{2} = \frac{10 \pm 4\sqrt{6}}{2} = 5 \pm 2\sqrt{6} \]

Since \( \left( \sqrt{3} + \sqrt{2} \right)^x \) must be positive, we have \( t = 5 + 2\sqrt{6} \).

The possible values for \( x \) are:

\[ x = 2 \quad \text{or} \quad x = -2 \]

The total number of solutions is 2.