If m and n are the roots of the quadratic equation `x^(2) + px + 8=0` with m- n = 2, then the value of p' is

Options

A. `pm8`
B. `pm 7`
C. `pm 6`
D. `pm 5`

Answer

To solve the problem step by step, we start with the given quadratic equation and the conditions provided.

### Step 1: Identify the roots and their relationships
We are given that \( m \) and \( n \) are the roots of the quadratic equation:
\[ x^2 + px + 8 = 0 \]
We also know that:
\[ m - n = 2 \]

### Step 2: Use Vieta's formulas
From Vieta's formulas, we know:
- The sum of the roots \( m + n = -p \)
- The product of the roots \( mn = 8 \)

### Step 3: Express \( n \) in terms of \( m \)
From the equation \( m - n = 2 \), we can express \( n \) as:
\[ n = m - 2 \]

### Step 4: Substitute \( n \) into the product equation
Now, substituting \( n \) into the product equation \( mn = 8 \):
\[ m(m - 2) = 8 \]
This simplifies to:
\[ m^2 - 2m - 8 = 0 \]

### Step 5: Solve the quadratic equation for \( m \)
We can solve the quadratic equation \( m^2 - 2m - 8 = 0 \) using the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -2 \), and \( c = -8 \):
\[ m = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \]
\[ m = \frac{2 \pm \sqrt{4 + 32}}{2} \]
\[ m = \frac{2 \pm \sqrt{36}}{2} \]
\[ m = \frac{2 \pm 6}{2} \]
This gives us two possible values for \( m \):
\[ m = \frac{8}{2} = 4 \quad \text{or} \quad m = \frac{-4}{2} = -2 \]

### Step 6: Find corresponding values of \( n \)
Now we find the corresponding values of \( n \):
1. If \( m = 4 \):
\[ n = 4 - 2 = 2 \]
2. If \( m = -2 \):
\[ n = -2 - 2 = -4 \]

### Step 7: Calculate \( p \)
Now, we can find \( p \) using \( m + n = -p \):
1. For \( m = 4 \) and \( n = 2 \):
\[ m + n = 4 + 2 = 6 \quad \Rightarrow \quad p = -6 \]
2. For \( m = -2 \) and \( n = -4 \):
\[ m + n = -2 - 4 = -6 \quad \Rightarrow \quad p = 6 \]

### Conclusion
Thus, the value of \( p \) can be either \( 6 \) or \( -6 \). Therefore, the final answer is:
\[ p = \pm 6 \]