Question

If the difference between the roots of $x^2+ax+b=0$ equals the difference between the roots of $x^2+bx+a=0,a\ne b$, then

• A. $a+b+4=0$
• B. $a-b+4=0$
• C. $a+b-4=0$
• D. $a-b-4=0$

Answer 1

The correct option is A $a+b+4=0$

$\alpha ,\beta berootsof{x}^2+ax+b=0\Rightarrow |\alpha -\beta |=\sqrt{a^2-4b}$
$\gamma ,\delta berootsof{x}^2+bx+a=0\Rightarrow |\gamma -\delta |=\sqrt{b^2-4a}$
$\therefore \sqrt{a^2-4b}=\sqrt{b^2-4a}$
$=a^2-4b=b^2-4a$
$=(a-b)(a+b)=-4(a-b)\Rightarrow a+b+4=0$