The sum of the real roots of the equation `|x-2|^(2)+|x-2|-2=0` is

Options

A. `2`
B. `3`
C. `4`
D. `10`

Answer

To solve the equation \( |x-2|^2 + |x-2| - 2 = 0 \), we will follow these steps:

### Step 1: Substitute \( t = |x - 2| \)
Let \( t = |x - 2| \). Then the equation becomes:
\[ t^2 + t - 2 = 0 \]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:
\[ t^2 + t - 2 = (t - 1)(t + 2) = 0 \]
This gives us the solutions:
\[ t - 1 = 0 \quad \text{or} \quad t + 2 = 0 \]
Thus, we have:
\[ t = 1 \quad \text{or} \quad t = -2 \]

### Step 3: Analyze the solutions for \( t \)
Since \( t = |x - 2| \) must be non-negative, we discard \( t = -2 \). Therefore, we only consider:
\[ t = 1 \]

### Step 4: Solve for \( x \)
Now, we substitute back to find \( x \):
1. If \( |x - 2| = 1 \), then we have two cases:
- Case 1: \( x - 2 = 1 \)
\[ x = 3 \]
- Case 2: \( x - 2 = -1 \)
\[ x = 1 \]

### Step 5: Find the sum of the roots
The real roots we found are \( x = 3 \) and \( x = 1 \). Therefore, the sum of the real roots is:
\[ 3 + 1 = 4 \]

### Final Answer
The sum of the real roots of the equation is \( \boxed{4} \).

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