Difference between the corresponding roots of x 2 + a x + b = 0   &   x 2 + b x + a = 0 is…

Let $\alpha , \beta \& \gamma , \delta$ are the roots of the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ respectively.

Therefore,

$\alpha +\beta =-a$

$\alpha \beta =b$

Also, for the second equation

$\gamma +\delta =-b$

 $\gamma \delta = a$

Given that,

$\alpha -\beta =\gamma -\delta$

$\Rightarrow {\left(\alpha -\beta \right)}^2={\left(\gamma -\delta \right)}^2$

$\Rightarrow {\left(\alpha +\beta \right)}^2-4\alpha \beta ={\left(\gamma +\delta \right)}^2-4\gamma \delta$

$\Rightarrow a^2-4b=b^2-4a$

$\Rightarrow \left(a^2-b^2\right)+4\left(a-b\right)=0$

$\Rightarrow \left(a-b\right)\left(a+b\right)+4\left(a-b\right)=0$

$\Rightarrow \left(a-b\right)\left(4+a+b\right)=0$

$\Rightarrow a+b+4 = 0$

$\left[\because a\ne b\right]$