Question

Let $x_1,x_2$ be the roots of $a{x}^2+bx+c=0$ and $x_1.x_2<0$, $x_1+x_2$ is non - zero. Roots of $x_1(x-x_2{)}^2+x_2(x-x_1{)}^2=0$ are

• A. Negative
• B. Real and of opposite signs
• C. positive
• D. non real

Answer 1

The correct option is B

Real and of opposite signs

Given $x_1(x-x_2{)}^2+x_2(x-x_1{)}^2=0$
$\Rightarrow x^2(x_1+x_2)-4x{x}_1{x}_2+x_1{x}_2(x_1+x_2)=0\Delta =16x_1^2{x}_2^2-4x_1{x}_2(x_1+x_2{)}^2>0(\because x_1{x}_2<0given)$
$\therefore$ Real roots
Also product of roots $x_1{x}_2<0$
$\therefore$ Roots are of opposite sign