Question

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

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

Answer 1

The correct option is A

$1:2:3$

Explanation for correct option

We know that if $\alpha ,\beta$ are the roots of the quadratic equation $p{x}^2+qx+r=0$ then

$\alpha +\beta =-\frac{q}{p}$

$\alpha \beta =\frac{r}{p}$

It is given that the roots of the equation $x^2+2x+3=0$ and $a{x}^2+bx+c=0$ are equal.

$\because$ The roots of the quadratic equation is equal so the coefficient of both the equation will be equal.

So comparing we get $a=1,b=2$ and $c=3$

Therefore the ratio of $a:b:c$ is $1:2:3$

Hence, option (A) is correct i.e. $1:2:3$