Question

If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0,a,b,c\in R$, have a common root, then $a:b:c$ is

• A. $3:2:1$
• B. $1:3:2$
• C. $3:1:2$
• D. $1:2:3$

Answer 1

The correct option is D $1:2:3$

For equation $x^2+2x+3=0$

$\Delta =2^2-4\left(1\right)\left(3\right)=4-12=-8<0$ so both roots are imaginary .

Hence, the roots are non-real. They will exist in complex conjugate pairs.

As one of the roots is common to $a{x}^2+bx+c=0$ , the other root will also be the complex conjugate of it.

Hence, the roots of the two equations will be the same.

Since $a,b,c\in R$.

If one root is common, then both roots are common .

Hence, $\frac{a}{1}=\frac{b}{2}=\frac{c}{3}$

$a:b:c=1:2:3$