if the difference of the roots of the equation ` x^(2)+ ax +b=0` is equal to the difference of the roots of the equation ` x^(2) +bx +a =0` ,then

Options

A. a + b =4
B. a = b = - 4
C. a -b = 4
D. a -b = - 4

Answer

To solve the problem, we need to analyze the given equations and derive the necessary conditions based on the difference of their roots.

### Step-by-Step Solution:

1. **Identify the equations**:
We have two quadratic equations:
\( x^2 + ax + b = 0 \)
\( x^2 + bx + a = 0 \)

2. **Roots of the first equation**:
Let the roots of the first equation be \( \alpha \) and \( \beta \).
By Vieta's formulas, we know:
- Sum of roots: \( \alpha + \beta = -a \)
- Product of roots: \( \alpha \beta = b \)

3. **Difference of the roots for the first equation**:
The difference of the roots can be expressed as:
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting the values from Vieta's formulas:
\[ \alpha - \beta = \sqrt{(-a)^2 - 4b} = \sqrt{a^2 - 4b} \]

4. **Roots of the second equation**:
Let the roots of the second equation be \( \gamma \) and \( \delta \).
Again, by Vieta's formulas:
- Sum of roots: \( \gamma + \delta = -b \)
- Product of roots: \( \gamma \delta = a \)

5. **Difference of the roots for the second equation**:
Similarly, the difference of the roots can be expressed as:
\[ \gamma - \delta = \sqrt{(\gamma + \delta)^2 - 4\gamma\delta} \]
Substituting the values from Vieta's formulas:
\[ \gamma - \delta = \sqrt{(-b)^2 - 4a} = \sqrt{b^2 - 4a} \]

6. **Setting the differences equal**:
According to the problem, the difference of the roots of both equations is equal:
\[ \sqrt{a^2 - 4b} = \sqrt{b^2 - 4a} \]

7. **Squaring both sides**:
To eliminate the square roots, we square both sides:
\[ a^2 - 4b = b^2 - 4a \]

8. **Rearranging the equation**:
Rearranging gives us:
\[ a^2 + 4a = b^2 + 4b \]

9. **Factoring**:
This can be factored as:
\[ a^2 - b^2 + 4a - 4b = 0 \]
Which can be rewritten using the difference of squares:
\[ (a - b)(a + b) + 4(a - b) = 0 \]
Factoring out \( (a - b) \):
\[ (a - b)(a + b + 4) = 0 \]

10. **Finding the conditions**:
This gives us two possible conditions:
- \( a - b = 0 \) (which implies \( a = b \))
- \( a + b + 4 = 0 \) (which implies \( a + b = -4 \))

### Conclusion:
The conditions derived from the problem are:
1. \( a = b \)
2. \( a + b = -4 \)

However, since \( a = b \) leads to both equations being identical, which is not a valid solution for the problem, we conclude that the valid condition is:
\[ \boxed{a + b = -4} \]