Let $S$ be the set of all real roots of the equation, $3^x(3^x - 1) + 2 = |3^x - 1| + |3^x - 2|$. Then $S$:

(1) contains exactly two elements.

(2) is a singleton.

(3) is an empty set.

(4) contains at least four elements.

Show Answer

Let $3^x = y$

$\therefore y(y - 1) + 2 = |y - 1| + |y - 2|$

Case 1: when $y > 2$: $y^2 - y + 2 = y - 1 + y - 2$; $y^2 - 3y + 5 = 0$; $D < 0$ [Equation not satisfy.]

Case 2: when $1 \leq y \leq 2$: $y^2 - y + 2 = y - 1 - y + 2$; $y^2 - y + 1 = 0$; $D < 0$ [Equation not satisfy.]

Case 3: when $y \leq 1$: $y^2 - y + 2 = -y + 1 - y + 2$; $y^2 + y - 1 = 0$

$\therefore y = \frac{-1 + \sqrt{5}}{2}$ [since Equation not Satisfy for negative]

$\therefore$ Only $-1 + \frac{\sqrt{5}}{2}$ satisfy equation