Let x1, x2 be the roots of ax2+bx+c=0 and x1.x2<0, x1+x2 is non - zero. Roots of x1(x−x2)2+x2(x−x1)2=0 are
Real and of opposite signs
Given x1(x−x2)2+x2(x−x1)2=0
⇒ x2(x1+x2)−4x x1 x2+x1 x2(x1+x2)=0Δ=16 x21x22−4x1x2(x1+x2)2>0 (∵ x1x2<0 given)
∴ Real roots
Also product of roots x1x2<0
∴ Roots are of opposite sign