Question

The roots of the equation $(b-c)x^2+2(c-a)x+(a-b)=0$ for $a,b,c\in R$ and $a\ne b\ne c$ are always

• A. real and distinct
• B. real and equal
• C. real
• D. imaginary

Answer 1

The correct option is A real and distinct

Let $f\left(x\right)=(b-c)x^2+2(c-a)x+a-b$
Consider $D=b^2-4ac$
$=2(c-a{)}^2-4(a-b\left)\right(b-c)$
$=4(a^2+b^2+c^2-ab-bc-ca)$
$=4\left(\frac{1}{2}\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2)$
$=2\left(\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2)\ge 0$
$\therefore$ Roots are real and distinct as $a\ne b\ne c$