If equations $$ax^2 + bx + c = 0$$, $$(a, b, c \in R, a \neq 0)$$ and $$2x^2 + 3x + 4 = 0$$ have a common root, then $$a : b : c$$ equals:

The given equations are $$ ax^2 + bx + c = 0 $$ (with $$ a, b, c $$ real numbers and $$ a \neq 0 $$) and $$ 2x^2 + 3x + 4 = 0 $$. They share a common root. To find the ratio $$ a : b : c $$, we start by examining the second equation.

The quadratic equation $$ 2x^2 + 3x + 4 = 0 $$ has a discriminant $$ D = b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot 4 = 9 - 32 = -23 $$, which is negative. Therefore, the roots are complex conjugates: $$ x = \frac{-3 \pm \sqrt{-23}}{4} = \frac{-3 \pm i\sqrt{23}}{4} $$.

Since the coefficients of both equations are real, if one complex root is common, its conjugate must also be a root of the first equation. A quadratic equation has at most two roots, so both roots must be common to both equations. This implies that the equations are proportional, meaning there exists a non-zero constant $$ k $$ such that:

$$ a = 2k, \quad b = 3k, \quad c = 4k $$

Thus, the ratio $$ a : b : c = 2k : 3k : 4k = 2 : 3 : 4 $$.

Now, comparing with the options:

• A. $$ 2 : 3 : 4 $$
• B. $$ 4 : 3 : 2 $$
• C. $$ 1 : 2 : 3 $$
• D. $$ 3 : 2 : 1 $$

The ratio $$ 2 : 3 : 4 $$ matches option A.

To verify, we can check if other options satisfy the condition of having a common root. Suppose we try option B: $$ a = 4k $$, $$ b = 3k $$, $$ c = 2k $$. The first equation becomes $$ 4k x^2 + 3k x + 2k = 0 $$, which simplifies to $$ 4x^2 + 3x + 2 = 0 $$ (since $$ k \neq 0 $$). The discriminant is $$ 3^2 - 4 \cdot 4 \cdot 2 = 9 - 32 = -23 $$, so roots are $$ \frac{-3 \pm i\sqrt{23}}{8} $$, which differ from the roots of $$ 2x^2 + 3x + 4 = 0 $$ ($$ \frac{-3 \pm i\sqrt{23}}{4} $$). Thus, no common root.

For option C: $$ a = k $$, $$ b = 2k $$, $$ c = 3k $$, the equation is $$ k x^2 + 2k x + 3k = 0 $$ or $$ x^2 + 2x + 3 = 0 $$. Discriminant is $$ 4 - 12 = -8 $$, roots $$ \frac{-2 \pm i\sqrt{8}}{2} = -1 \pm i\sqrt{2} $$, not matching.

For option D: $$ a = 3k $$, $$ b = 2k $$, $$ c = k $$, the equation is $$ 3k x^2 + 2k x + k = 0 $$ or $$ 3x^2 + 2x + 1 = 0 $$. Discriminant is $$ 4 - 12 = -8 $$, roots $$ \frac{-2 \pm i\sqrt{8}}{6} = \frac{-1 \pm i\sqrt{2}}{3} $$, not matching.

Only option A gives proportional equations with identical roots, ensuring a common root (in fact, both roots are common).

Hence, the correct answer is Option A.