Animated Solution for Mathematics - Quadratic Equations: 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 $S$:

The given equation is $3^x(3^x-1)+2=|3^x-1|+|3^x-2|$. At first glance, it appears to be a complex mix of exponentials and absolute values.

To simplify, we recognize that the term $3^x$ repeats throughout the expression. We perform the substitution $3^x=t$, which transforms the equation into:

This substitution strips away the complexity, revealing the underlying algebraic structure.

Before proceeding with the algebra, we must respect the domain constraints. Since $x$ is any real number, the exponential function $3^x$ is strictly positive.

Therefore, our new variable must satisfy the condition $t>0$. This constraint acts as a filter; any solution we find for $t$ that is less than or equal to zero must be discarded immediately.

We define the right-hand side as $g(t)=|t-1|+|t-2|$. This is a piecewise function with critical points at $t=1$ and $t=2$. We must analyze the equation across three distinct regions.

In this region, both $(t-1)$ and $(t-2)$ are non-positive. Thus, they emerge from the modulus with a sign flip: $g(t)=-(t-1)-(t-2)=3-2t$.

The equation becomes $t^2-t+2=3-2t$, which simplifies to the quadratic equation:

Using the quadratic formula, we find $t=\frac{-1\pm \sqrt{5}}{2}$. Since $t>0$, we accept the positive root:

This value satisfies $t\le 1$, confirming it as a valid solution.

In this interval, $(t-1)$ is positive, but $(t-2)$ is negative. The modulus simplifies to $g(t)=(t-1)-(t-2)=1$.

The equation becomes $t^2-t+2=1$, or $t^2-t+1=0$. Checking the discriminant:

Since $D<0$, there are no real roots in this interval.

Finally, for $t\ge 2$, both terms are positive. Thus, $g(t)=(t-1)+(t-2)=2t-3$.

The equation becomes $t^2-t+2=2t-3$, which simplifies to $t^2-3t+5=0$. The discriminant is:

Again, $D<0$, meaning no real roots exist in this interval.

We have systematically checked every interval and found exactly one valid value for $t$:

Because $3^x=t$ is a one-to-one function, this single value of $t$ corresponds to exactly one real value of $x$. We have navigated the traps and respected the domain constraints.

The set $S$ contains exactly one element. Therefore, $S$ is a singleton.