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 equal to
$\eqalign{
  & 1)1:2:3 \cr
  & 2)3:2:1 \cr
  & 3)1:3:2 \cr
  & 4)3:1:2 \cr} $

Answer 1

Hint: The given question contains an equation. We have to find out the ratio of the variables of the generalized equation. The condition is that they have a common root. So, we need to find whether the roots are imaginary or real by finding the discriminant. Then, we can find the common root and find out the ratio for the variables.
Formulas used to solve this problem is:
For an equation in the form $a{x^2} + bx + c = 0$,
Discriminant is given by the formula, $d = {b^2} - 4ac$

Complete step-by-step answer:
The given equation is as follows,
${x^2} + 2x + 3 = 0$
Now, let us find out the discriminant of the above equation.
$\eqalign{
  & \Rightarrow d = \left( {{2^2}} \right) - 4\left( 1 \right)\left( 3 \right) \cr
  & \Rightarrow d = 4 - 12 \cr
  & \Rightarrow d = - 8 \cr} $
$ - 8$is less than $0$. Therefore, this equation has imaginary roots. Both the roots are imaginary.
That means they are non-real. Hence, they will exist in complex conjugate pairs.
It is given that one of the roots is common to $a{x^2} + bx + c = 0$. One of the roots is imaginary. Hence, the other root will also be a complex conjugate of it.
It is given that, $a,b,c \in R$ and one root is common. Since one root is common, both the roots are common.
Therefore, the roots of both the given equations will be the same.
Hence, $\dfrac{a}{1} = \dfrac{b}{2} = \dfrac{c}{3}$
That is, $a:b:c = 1:2:3$
The final answer is $1:2:3$
Hence, option (1) is the correct answer.
So, the correct answer is “Option 1”.

Note: The question does not directly ask us to find the discriminant.
The key word is ‘common root’.
We need not find the actual roots either. We just need to compare the nature of roots of one equation with the other.