If the difference between the corresponding roots of the equations x^2 +ax+b=0 and x^2+bx+a=0 is the same , then find the value of a+b .

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

and   $\gamma ,\delta$  be the roots of  $x^{2}+bx+a=0$

Then  given

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

$\Rightarrow$                        $\sqrt{a^{2}-4b}=\sqrt{b^{2}-4a}$

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

$\Rightarrow$                 $(a^{2}-b^{2})+4(a-b)=0\Rightarrow (a-b)(a+b+4)=0$

$\because$                 $a-b\neq0$

$\therefore$                 $(a+b+4)=0\Rightarrow a+b=-4.$

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