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Basics & Logarithms: Sum Possible Values Equation

JEE Maths question with a full step-by-step solution.

Question

Sum of all possible values of

x

of the equation

(\sqrt{3}+\sqrt{2})^x + (\sqrt{3}-\sqrt{2})^x - 2\sqrt{3} = 0

is:

2

0

correct

-1

\dfrac{1}{2}

Solution

Step 1: Use the reciprocal relationship Since

(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2}) = 1

, let

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

. Then

(\sqrt{3}-\sqrt{2})^x = \dfrac{1}{t}

. Step 2: Form a quadratic in

t

t + \frac{1}{t} = 2\sqrt{3} \implies t^2 - 2\sqrt{3}\,t + 1 = 0

t = \frac{2\sqrt{3} \pm \sqrt{12-4}}{2} = \sqrt{3} \pm \sqrt{2}

Step 3: Solve for

x

(\sqrt{3}+\sqrt{2})^x = \sqrt{3}+\sqrt{2} \implies x = 1

(\sqrt{3}+\sqrt{2})^x = \sqrt{3}-\sqrt{2} = (\sqrt{3}+\sqrt{2})^{-1} \implies x = -1

Step 4: Sum the solutions

1 + (-1) = 0

Answer: (2)

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