If m and n are roots of the equation (x+p)(x+q)-k=0 then find the roots of the equation (x-m)(x-n)+k=0

Options

A. `p and q`
B. `1/p and 1/q`
C. `-p and -q`
D. `p+q and p-q`

Explanation

To solve the problem step by step, we start with the given equation and its roots.

### Step 1: Write the given equation
The given equation is:
\[ (x + p)(x + q) - k = 0 \]

### Step 2: Expand the equation
Expanding the left-hand side, we have:
\[ x^2 + (p + q)x + (pq - k) = 0 \]

### Step 3: Identify the roots
From the equation \(x^2 + (p + q)x + (pq - k) = 0\), we know that the roots are \(m\) and \(n\). By Vieta's formulas:
- The sum of the roots \(m + n = -(p + q)\)
- The product of the roots \(mn = pq - k\)

### Step 4: Write the new equation
We need to find the roots of the equation:
\[ (x - m)(x - n) + k = 0 \]

### Step 5: Expand the new equation
Expanding this equation gives:
\[ x^2 - (m + n)x + mn + k = 0 \]

### Step 6: Substitute the values of \(m + n\) and \(mn\)
Using the values we found from the first equation:
- Substitute \(m + n = -(p + q)\)
- Substitute \(mn = pq - k\)

The new equation becomes:
\[ x^2 - (-(p + q))x + (pq - k + k) = 0 \]
This simplifies to:
\[ x^2 + (p + q)x + pq = 0 \]

### Step 7: Factor the new equation
Now, we can factor this equation:
\[ (x + p)(x + q) = 0 \]

### Step 8: Find the roots
Setting each factor to zero gives us the roots:
\[ x + p = 0 \quad \Rightarrow \quad x = -p \]
\[ x + q = 0 \quad \Rightarrow \quad x = -q \]

Thus, the roots of the equation \((x - m)(x - n) + k = 0\) are:
\[ \boxed{-p \text{ and } -q} \]