The sum of all the roots of the equation `|x^2-8x +15|-2x +7 +0` is

Options

A. `9-sqrt3`
B. `9+sqrt3`
C. `11 -sqrt3`
D. `11 + sqrt3`

Answer

To solve the equation \( |x^2 - 8x + 15| - 2x + 7 = 0 \), we will break it down into two cases based on the definition of the absolute value.

### Step 1: Rewrite the equation
We start with the equation:
\[ |x^2 - 8x + 15| - 2x + 7 = 0 \]
This can be rewritten as:
\[ |x^2 - 8x + 15| = 2x - 7 \]

### Step 2: Factor the quadratic expression
The expression inside the absolute value can be factored:
\[ x^2 - 8x + 15 = (x - 3)(x - 5) \]
Thus, we have:
\[ |(x - 3)(x - 5)| = 2x - 7 \]

### Step 3: Consider Case 1: \(x < 3\) or \(x > 5\)
In this case, \(|(x - 3)(x - 5)| = (x - 3)(x - 5)\). Therefore, the equation becomes:
\[ (x - 3)(x - 5) = 2x - 7 \]
Expanding the left side:
\[ x^2 - 8x + 15 = 2x - 7 \]
Rearranging gives:
\[ x^2 - 10x + 22 = 0 \]

### Step 4: Solve the quadratic equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 22}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{10 \pm \sqrt{100 - 88}}{2} = \frac{10 \pm \sqrt{12}}{2} = \frac{10 \pm 2\sqrt{3}}{2} = 5 \pm \sqrt{3} \]
The roots are \(x = 5 + \sqrt{3}\) and \(x = 5 - \sqrt{3}\). Both roots are valid since \(5 + \sqrt{3} > 5\) and \(5 - \sqrt{3} < 5\).

### Step 5: Consider Case 2: \(3 \leq x \leq 5\)
In this case, \(|(x - 3)(x - 5)| = -(x - 3)(x - 5)\). Therefore, the equation becomes:
\[ -(x - 3)(x - 5) = 2x - 7 \]
Expanding gives:
\[ -(x^2 - 8x + 15) = 2x - 7 \]
Rearranging gives:
\[ -x^2 + 8x - 15 = 2x - 7 \]
Which simplifies to:
\[ -x^2 + 6x - 8 = 0 \quad \Rightarrow \quad x^2 - 6x + 8 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula again:
\[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{6 \pm \sqrt{36 - 32}}{2} = \frac{6 \pm \sqrt{4}}{2} = \frac{6 \pm 2}{2} \]
The roots are:
\[ x = \frac{8}{2} = 4 \quad \text{and} \quad x = \frac{4}{2} = 2 \]
Only \(x = 4\) is valid in the interval \(3 \leq x \leq 5\).

### Step 7: Sum of all roots
The valid roots we found are:
1. \(5 + \sqrt{3}\)
2. \(5 - \sqrt{3}\)
3. \(4\)

Now, we calculate the sum:
\[ (5 + \sqrt{3}) + (5 - \sqrt{3}) + 4 = 5 + 5 + 4 = 14 \]

### Final Answer
The sum of all the roots of the equation is:
\[ \boxed{14} \]