If $a \ne b$ and difference between the roots of the equations $x^2+ax+b=0\text{ and }{x}^2+bx+a=0\text{ is the }$same, then 

$\text{ Let }\alpha ,\beta \text{ be the roots of }{x}^2+ax+b=0\text{ and }\gamma \text{,}$

$\delta \text{ be the roots of }{x}^2+bx+a=0\text{. }$

We are given

$\begin{array}{l} |\alpha -\beta |=|\gamma -\delta |\\ \Rightarrow |\alpha -\beta {|}^2=|\gamma -\delta {|}^2\\ \Rightarrow (\alpha +\beta {)}^2-4\alpha \beta =(\gamma +\delta {)}^2-4\gamma \delta \\ \Rightarrow a^2-4b=b^2-4a\\ \Rightarrow a^2-b^2+4(a-b)=0\\ \Rightarrow (a-b)(a+b+4)=0\\ \text{ As }a\ne b,a+b+4=0.\end{array}$