The sum of al real roots of the equation `|x-3|^(2) + |x-3|-2=0` is :

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the equation \( |x-3|^2 + |x-3| - 2 = 0 \), we will follow these steps:

### Step 1: Substitute \( |x-3| \) with a variable
Let \( t = |x-3| \). Then, the equation becomes:
\[ t^2 + t - 2 = 0 \]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:
\[ t^2 + t - 2 = (t + 2)(t - 1) = 0 \]

### Step 3: Solve for \( t \)
Setting each factor to zero gives us:
1. \( t + 2 = 0 \) → \( t = -2 \)
2. \( t - 1 = 0 \) → \( t = 1 \)

### Step 4: Analyze the solutions for \( t \)
Since \( t = |x-3| \) cannot be negative, we discard \( t = -2 \). Therefore, we only consider:
\[ t = 1 \]

### Step 5: Solve for \( x \)
Now we revert back to \( |x-3| = 1 \). This gives us two equations:
1. \( x - 3 = 1 \) → \( x = 4 \)
2. \( x - 3 = -1 \) → \( x = 2 \)

### Step 6: Find the sum of all real roots
The real roots we found are \( x = 4 \) and \( x = 2 \). Therefore, the sum of all real roots is:
\[ 4 + 2 = 6 \]

### Final Answer
The sum of all real roots of the equation is \( 6 \).

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