Question

If $a+b+c=0$ and $a,b,c$ are rational, then the roots of the equation

$(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$ are

• A. Rational
• B. Irrational
• C. Imaginary
• D. Equal

Answer 1

The correct option is A Rational

We have, $(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$

$D={(c+a-b)}^2-4(b+c-a)(a+b-c)={(a+b+c-2b)}^2-4(a+b+c-2a)(a+b+c-2c)={(-2b)}^2-4(-2a)(-2c)=4(b^2-4ac)=4({(-a-c)}^2-4ac)=4{(a-c)}^2={\left(2(a-c)\right)}^2$

which is a perfect square.

Hence roots are rational.