The real numbers α,β  are of opposite signs and |α|≠|β|.  The roots of α(x−β)2+β(x−α)2 =0  are

The given equation is α(x−β)2+β(x−α)2 =0 ⇒α(x2+β2−2xαβ)+β(x2+α2−2αx) ⇒x2(α+β)  −4αβx+αβ(α+β)=0 Or  x2−  (4αβα+β)x+αβ=0 Here, coefficient of x2 and the constant term are of opposite signs (∵  αβ<0) Therefore, the roots of the given equation are real and of opposite signs.