The sum of the roots of equation `(x-4)^(2)-8|x-4|+15=0` is

Options

A. `-8`
B. `16`
C. `8`
D. `0`

Answer

To solve the equation \((x-4)^{2}-8|x-4|+15=0\) and find the sum of its roots, we can follow these steps:

### Step 1: Substitute \(t = x - 4\)
Let \(t = x - 4\). Then the equation becomes:
\[ t^2 - 8|t| + 15 = 0 \]

### Step 2: Consider two cases for \(|t|\)
Since \(|t|\) can be either \(t\) (when \(t \geq 0\)) or \(-t\) (when \(t < 0\)), we will consider these two cases separately.

#### Case 1: \(t \geq 0\)
In this case, \(|t| = t\). The equation simplifies to:
\[ t^2 - 8t + 15 = 0 \]
We can factor this quadratic equation:
\[ (t - 5)(t - 3) = 0 \]
Thus, the solutions for \(t\) are:
\[ t = 5 \quad \text{or} \quad t = 3 \]

### Step 3: Find corresponding \(x\) values for Case 1
Substituting back for \(x\):
1. If \(t = 5\), then \(x - 4 = 5 \Rightarrow x = 9\)
2. If \(t = 3\), then \(x - 4 = 3 \Rightarrow x = 7\)

### Step 4: Case 2: \(t < 0\)
In this case, \(|t| = -t\). The equation becomes:
\[ t^2 + 8t + 15 = 0 \]
We can factor this quadratic equation:
\[ (t + 5)(t + 3) = 0 \]
Thus, the solutions for \(t\) are:
\[ t = -5 \quad \text{or} \quad t = -3 \]

### Step 5: Find corresponding \(x\) values for Case 2
Substituting back for \(x\):
1. If \(t = -5\), then \(x - 4 = -5 \Rightarrow x = -1\)
2. If \(t = -3\), then \(x - 4 = -3 \Rightarrow x = 1\)

### Step 6: Sum of all roots
Now we have the roots:
- From Case 1: \(x = 9\) and \(x = 7\)
- From Case 2: \(x = -1\) and \(x = 1\)

The sum of the roots is:
\[ 9 + 7 + (-1) + 1 = 16 \]

### Final Answer
The sum of the roots of the equation is:
\[ \boxed{16} \]