Question

If $a,b,c$ are rational and $a\ne b,b\ne a+c,$ then the roots of the equation $(a+c-b)x^2+2cx+(b+c-a)=0$ are

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

Answer 1

The correct option is C rational and distinct

$D=(2c{)}^2-4(a+c-b\left)\right(b+c-a)$
$\Rightarrow D=4c^2-4(c+(a-b\left)\right)(c-(a-b\left)\right)$
$=4c^2-4(c^2-(a-b{)}^2)$
$\Rightarrow D=4(a-b{)}^2$
$\Rightarrow D>0(\because a\ne b)$ and is a perfect square.
So, roots are rational and distinct.