Question

The equations $a{x}^2+bx+c=0$ and $x^2+x+c=0$ have
( where $c\ge 1$ )

• A. a common root when $a,b\in R$
• B. two common roots when $a,b\in R$
• C. a common root only when $a=b\ne c$ and $a,b\in R$
• D. two common roots only when $a=b=c$

Answer 1

The correct option is B two common roots when $a,b\in R$

$x^2+x+c=0$
$\Delta =1-4c<0(\because c\ge 1)$
$\Delta <0$, hence roots are non-real.
We know that for a quadratic equation with real coefficients, non-real roots always occur in conjugate pair.

If $a,b,c\in R$,
the two quadratic equations with real coefficients cannot have only one non-real root in common.
So, both the roots are common.