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

Answer: (C)

2

Answer 2

Consider:

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

Let:

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

Thus:

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

Substitute and simplify:

$t + \frac{1}{t} = 10$

Multiplying through by $t$ gives:

$t^2 - 10t + 1 = 0$

Solving this quadratic equation:

$t = \frac{10 \pm \sqrt{100 - 4}}{2} = \frac{10 \pm \sqrt{96}}{2} = \frac{10 \pm 4\sqrt{6}}{2} = 5 \pm 2\sqrt{6}$

Since:

$\left( \sqrt{3} + \sqrt{2} \right)^x > 0, \quad t = 5 + 2\sqrt{6}$

Thus, the corresponding values of $x$ are:
$x = 2 \quad \text{or} \quad x = -2$

Number of solutions = 2.