If the equations $$x^2+ax+b=0$$​ and $$x^2+bx+a=0$$​ have a common root, then find the value of $$a+b$$​ (where a is not equal to b)

Correct option is D

Given:
Two quadratic equations:

​$$x^2 + ax + b = 0 \\x^2 + bx + a = 0$$​​

Concept Used:

If two quadratic equations f(x) = 0 and g(x) = 0 have a common root $$\alpha$$​, then:
​$$f(\alpha) = 0 \quad \text{and} \quad g(\alpha) = 0$$​
Solution:
Let $$\alpha$$​ be the common root. Then:
​$$\alpha^2 + a\alpha + b = 0 \quad \text{(1)} \\\alpha^2 + b\alpha + a = 0 \quad \text{(2)}$$​​
Subtract (2) from (1):

​$$(a - b)\alpha + (b - a) = 0 \\(a - b)(\alpha - 1) = 0$$​​
Since $$a \neq b$$​ , we must have:
​$$\alpha - 1 = 0 \\ \alpha = 1$$​​
Using equation (1):
​$$1^2 + a(1) + b = 0$$​​

1 + a + b = 0

a + b = -1