Key Concepts and Formulas

• Discriminant of a Quadratic Equation: For a quadratic equation $Ax^2 + Bx + C = 0$, the discriminant is given by $D = B^2 - 4AC$. If $D < 0$, the equation has complex roots.
• Conjugate Root Theorem: If a polynomial equation with real coefficients has a complex root $p + iq$, then its conjugate $p - iq$ is also a root.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha \beta = \frac{c}{a}$.

Step-by-Step Solution

Step 1: Determine the Nature of the Roots of the First Equation

We analyze the equation $x^2 + 2x + 3 = 0$ to determine if its roots are real or complex. We calculate the discriminant to determine the nature of the roots.

The discriminant $D$ is given by: $D = B^2 - 4AC = (2)^2 - 4(1)(3) = 4 - 12 = -8$

Since $D = -8 < 0$, the roots of the equation $x^2 + 2x + 3 = 0$ are complex (imaginary). Let the roots be $\alpha$ and $\beta$.

Step 2: Apply the Conjugate Root Theorem

The problem states that the second equation, $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$, has a common root with the first equation. Since the first equation has complex roots and the second equation has real coefficients, the Conjugate Root Theorem applies. This means that if $\alpha$ is a common root, its complex conjugate $\overline{\alpha}$ must also be a root of both equations. Therefore, both roots of $x^2 + 2x + 3 = 0$ are also roots of $ax^2 + bx + c = 0$.

Step 3: Relate the Coefficients of the Two Equations

Since both quadratic equations have the same roots, their coefficients must be proportional. This means there exists a constant $k$ such that $a = k(1)$, $b = k(2)$, and $c = k(3)$. We can express this proportionality as ratios.

We can also use Vieta's formulas. For the first equation, $x^2 + 2x + 3 = 0$, we have: Sum of roots: $\alpha + \beta = -\frac{2}{1} = -2$ Product of roots: $\alpha \beta = \frac{3}{1} = 3$

For the second equation, $ax^2 + bx + c = 0$, we have: Sum of roots: $\alpha + \beta = -\frac{b}{a}$ Product of roots: $\alpha \beta = \frac{c}{a}$

Equating the sum and product of roots, we get: $-\frac{b}{a} = -2 \Rightarrow b = 2a$ $\frac{c}{a} = 3 \Rightarrow c = 3a$

Therefore, the ratio $a:b:c$ is: $a:b:c = a:2a:3a$

Dividing by $a$ (assuming $a \neq 0$), we get: $a:b:c = 1:2:3$

Common Mistakes & Tips

• Importance of Real Coefficients: The Conjugate Root Theorem only applies when the coefficients of the polynomial are real.
• Check Discriminant: Always verify that the first equation has complex roots before applying the Conjugate Root Theorem.
• Proportionality: Remember that the coefficients are proportional, not necessarily equal.

Summary

We analyzed the discriminant of the first quadratic equation to determine that its roots are complex. Because the second quadratic equation has real coefficients and shares a root with the first equation, it must share both roots due to the Conjugate Root Theorem. This implies that the coefficients of the two quadratic equations are proportional, leading to the ratio $a:b:c = 1:2:3$.

The final answer is \boxed{1:2:3}, which corresponds to option (A).