Let S be the set of all real roots of the equation `4^x (4^x -1)+2=|4^x -1|+|4^x-2)`.Then S:

A. is an empty set
B. contains at least four elements
C. contains exactly two elements
D. is a singleton

Options

A. is an empty set
B. contains at least four elements
C. contains exactly two elements
D. is a singleton

Explanation

To solve the equation \( 4^x (4^x - 1) + 2 = |4^x - 1| + |4^x - 2| \), we will follow these steps:

### Step 1: Substitute \( 4^x \) with \( t \)
Let \( t = 4^x \). The equation then becomes:
\[ t(t - 1) + 2 = |t - 1| + |t - 2| \]
This simplifies to:
\[ t^2 - t + 2 = |t - 1| + |t - 2| \]

### Step 2: Analyze the absolute values
We need to consider different cases based on the value of \( t \) because of the absolute value expressions.

#### Case 1: \( t < 1 \)
In this case, both \( |t - 1| \) and \( |t - 2| \) will be negative:
\[ |t - 1| = 1 - t \quad \text{and} \quad |t - 2| = 2 - t \]
Thus, the equation becomes:
\[ t^2 - t + 2 = (1 - t) + (2 - t) \]
Simplifying this gives:
\[ t^2 - t + 2 = 3 - 2t \]
Rearranging leads to:
\[ t^2 + t - 1 = 0 \]

### Step 3: Solve the quadratic equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{5}}{2} \]
This gives two potential solutions:
\[ t_1 = \frac{-1 + \sqrt{5}}{2}, \quad t_2 = \frac{-1 - \sqrt{5}}{2} \]
Since \( t = 4^x \) must be positive, we discard \( t_2 \) because it is negative. Thus, we have one valid solution from this case:
\[ t = \frac{-1 + \sqrt{5}}{2} \]

### Step 4: Case 2: \( 1 \leq t < 2 \)
Here, \( |t - 1| = t - 1 \) and \( |t - 2| = 2 - t \):
\[ t^2 - t + 2 = (t - 1) + (2 - t) \]
This simplifies to:
\[ t^2 - t + 2 = 1 \]
Rearranging gives:
\[ t^2 - t + 1 = 0 \]
Calculating the discriminant:
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since the discriminant is negative, there are no solutions in this case.

### Step 5: Case 3: \( t \geq 2 \)
In this case, both absolute values are positive:
\[ |t - 1| = t - 1 \quad \text{and} \quad |t - 2| = t - 2 \]
Thus, the equation becomes:
\[ t^2 - t + 2 = (t - 1) + (t - 2) \]
This simplifies to:
\[ t^2 - t + 2 = 2t - 3 \]
Rearranging gives:
\[ t^2 - 3t + 5 = 0 \]
Calculating the discriminant:
\[ D = (-3)^2 - 4 \cdot 1 \cdot 5 = 9 - 20 = -11 \]
Again, the discriminant is negative, so there are no solutions in this case.

### Conclusion
The only solution we found is from Case 1:
\[ t = \frac{-1 + \sqrt{5}}{2} \]
Since \( t = 4^x \), we can find \( x \):
\[ 4^x = \frac{-1 + \sqrt{5}}{2} \]
Thus, the set \( S \) contains exactly one element.

### Final Answer
The set \( S \) is a singleton.