If the roots of the equation `x^(2)+px+q=0` differ from the roots of the equation `x^(2)+qx+p=0` by the same quantity, then the value of p+q is

Options

A. -1
B. -2
C. -4
D. none

Answer

To solve the problem, we need to determine the value of \( p + q \) given that the roots of the equations \( x^2 + px + q = 0 \) and \( x^2 + qx + p = 0 \) differ by the same quantity.

### Step-by-Step Solution:

1. **Define the Roots**:
Let the roots of the first equation \( x^2 + px + q = 0 \) be \( \alpha_1 \) and \( \beta_1 \).
By Vieta's formulas:
- Sum of roots: \( \alpha_1 + \beta_1 = -p \)
- Product of roots: \( \alpha_1 \beta_1 = q \)

2. **Define the Roots of the Second Equation**:
Let the roots of the second equation \( x^2 + qx + p = 0 \) be \( \alpha_2 \) and \( \beta_2 \).
Again, by Vieta's formulas:
- Sum of roots: \( \alpha_2 + \beta_2 = -q \)
- Product of roots: \( \alpha_2 \beta_2 = p \)

3. **Express the Difference of Roots**:
According to the problem, the roots differ by the same quantity. Thus, we can express this as:
\[ \alpha_1 - \beta_1 = k \quad \text{and} \quad \alpha_2 - \beta_2 = k \]
for some constant \( k \).

4. **Square the Differences**:
We can express the square of the differences:
\[ (\alpha_1 - \beta_1)^2 = (\alpha_1 + \beta_1)^2 - 4\alpha_1\beta_1 \]
This gives us:
\[ k^2 = (-p)^2 - 4q \quad \Rightarrow \quad k^2 = p^2 - 4q \]

Similarly, for the second equation:
\[ (\alpha_2 - \beta_2)^2 = (\alpha_2 + \beta_2)^2 - 4\alpha_2\beta_2 \]
This gives:
\[ k^2 = (-q)^2 - 4p \quad \Rightarrow \quad k^2 = q^2 - 4p \]

5. **Set the Two Expressions for \( k^2 \) Equal**:
Since both expressions for \( k^2 \) are equal, we have:
\[ p^2 - 4q = q^2 - 4p \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ p^2 - q^2 = 4q - 4p \]
This can be factored as:
\[ (p - q)(p + q) = 4(q - p) \]
Rearranging further, we get:
\[ (p - q)(p + q + 4) = 0 \]

7. **Finding Possible Solutions**:
This equation implies two cases:
- Case 1: \( p - q = 0 \) (i.e., \( p = q \))
- Case 2: \( p + q + 4 = 0 \) (i.e., \( p + q = -4 \))

8. **Conclusion**:
Since we are looking for \( p + q \), we find that:
\[ p + q = -4 \]

Thus, the value of \( p + q \) is \( \boxed{-4} \).