If `f:(2,-oo) -> [8, oo)` is a surjective function defined by `f(x) = x^2 - (p-2)x + 3p-2,p in R` then sum of values of p is `m+sqrtn`, where `m, n in N`. Find the value of `n/m`.

Explanation

To solve the problem step by step, we will analyze the given function and its conditions.

### Step 1: Understand the Function
The function is given as:
\[ f(x) = x^2 - (p-2)x + (3p-2) \]
with the domain \( x \in (2, -\infty) \) and the range \( f(x) \in [8, \infty) \).

### Step 2: Identify the Nature of the Function
Since \( f(x) \) is a quadratic function, it opens upwards (as the coefficient of \( x^2 \) is positive). The minimum value of this function occurs at the vertex.

### Step 3: Find the Vertex
The x-coordinate of the vertex of a quadratic function \( ax^2 + bx + c \) is given by:
\[ x_v = -\frac{b}{2a} \]
For our function:
- \( a = 1 \)
- \( b = -(p-2) \)

Thus, the x-coordinate of the vertex is:
\[ x_v = \frac{p-2}{2} \]

### Step 4: Determine the Minimum Value
The y-coordinate of the vertex (minimum value of \( f(x) \)) can be calculated by substituting \( x_v \) back into the function:
\[ f(x_v) = f\left(\frac{p-2}{2}\right) \]

Calculating this gives:
\[ f\left(\frac{p-2}{2}\right) = \left(\frac{p-2}{2}\right)^2 - (p-2)\left(\frac{p-2}{2}\right) + (3p-2) \]
\[ = \frac{(p-2)^2}{4} - \frac{(p-2)^2}{2} + (3p-2) \]
\[ = \frac{(p-2)^2}{4} - \frac{2(p-2)^2}{4} + (3p-2) \]
\[ = -\frac{(p-2)^2}{4} + (3p-2) \]
\[ = 3p - 2 - \frac{(p-2)^2}{4} \]

### Step 5: Set Minimum Value Condition
For the function to be surjective (onto), the minimum value must be at least 8:
\[ 3p - 2 - \frac{(p-2)^2}{4} \geq 8 \]
Rearranging gives:
\[ 3p - 10 \geq \frac{(p-2)^2}{4} \]
Multiplying through by 4 to eliminate the fraction:
\[ 12p - 40 \geq (p-2)^2 \]
Expanding the right side:
\[ 12p - 40 \geq p^2 - 4p + 4 \]
Rearranging gives:
\[ p^2 - 16p + 44 \leq 0 \]

### Step 6: Solve the Quadratic Inequality
The roots of the equation \( p^2 - 16p + 44 = 0 \) can be found using the quadratic formula:
\[ p = \frac{16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot 44}}{2 \cdot 1} \]
\[ = \frac{16 \pm \sqrt{256 - 176}}{2} \]
\[ = \frac{16 \pm \sqrt{80}}{2} \]
\[ = \frac{16 \pm 4\sqrt{5}}{2} \]
\[ = 8 \pm 2\sqrt{5} \]

### Step 7: Find the Sum of Values of \( p \)
The values of \( p \) are \( 8 + 2\sqrt{5} \) and \( 8 - 2\sqrt{5} \). The sum is:
\[ (8 + 2\sqrt{5}) + (8 - 2\sqrt{5}) = 16 \]

### Step 8: Express the Sum in the Required Form
We need to express the sum as \( m + \sqrt{n} \):
\[ 16 = 16 + \sqrt{0} \]
Thus, \( m = 16 \) and \( n = 0 \).

### Step 9: Calculate \( \frac{n}{m} \)
Now, we find:
\[ \frac{n}{m} = \frac{0}{16} = 0 \]

### Final Answer
The value of \( \frac{n}{m} \) is:
\[ \boxed{0} \]