Question

If $a,b,c$ are distinct non-zero rational numbers such that $a+b+c=0$, then both the roots of the equation $(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$ are

• A. rational and unequal
• B. irrational
• C. non-real
• D. equal

Answer 1

The correct option is A rational and unequal

$(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$
As $a+b+c=0$, the equation becomes
$-2a{x}^2-2bx-2c=0$
$\Rightarrow a{x}^2+bx+c=0$

Putting $x=1$
$a+b+c=0$
So, $x=1$ is a root of the equation.
Let the other root be $\alpha$
Product of roots,
$1\times \alpha =\frac{c}{a}\Rightarrow \alpha =\frac{c}{a}$
As $a$ and $c$ both are rational numbers,
$\alpha$ is also rational.

Hence, both the roots are rational and unequal.