Question

The number of positive real roots of the equation \[ 3^{x+1} + 3^{-x+1} = 10 \] is:

• A. 3
• B. 2
• C. 1
• D. Infinitely many

Hint:

Try substitution like $y = a^x$ to reduce symmetric exponential equations.

Answer 1

The Correct Option is C

Let $y = 3^x$ Then $3^{x+1} = 3 \cdot y,\quad 3^{-x+1} = 3/y$ So equation becomes: \[ 3y + \frac{3}{y} = 10 \Rightarrow y + \frac{1}{y} = \frac{10}{3} \] Multiply both sides: \[ y^2 + 1 = \frac{10}{3}y \Rightarrow 3y^2 - 10y + 3 = 0 \Rightarrow \text{Discriminant } = 100 - 36 = 64>0 \Rightarrow \text{2 roots in } y>0 \] But only 1 positive $x$ gives positive $y$ ⇒ $\boxed{1}$ positive real root.