Question

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

• A. imaginary
• B. irrational
• C. rational and equal
• D. rational and distinct

Answer 1

The correct option is D rational and distinct

$(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$
Sum of coefficients $=0$
Since, the sum of coefficients is zero, so the given quadratic equation has one root to be $1.$

Let another root be $\alpha .$
Product of roots $=\frac{c+a-2b}{a+b-2c}$
$\Rightarrow 1\times \alpha =\frac{a+c-2b}{a+b-2c}$
$\Rightarrow \alpha =\frac{a+c-2b}{a+b-2c}$
$\therefore$ Roots are rational and distinct.