Q. If $ax^{2} + bx + c = 0$ and $2x^{2} + 3x + 4 = 0$ have a common root where $a,\ b,\ c \in N$ (set of natural numbers), the least value of $a + b + c$ is
A) 13
B) 11
C) 7
D) 9
Explanation
(d)
Let $\alpha$ be the common root of the given equations
Then, $a\alpha^{2} + b\alpha + c = 0$
And $2\alpha^{2} + 3\alpha + 4 = 0$
$\Rightarrow \ \ \ \ \alpha^{2} + (a - 2) + \alpha(b - 3) + c - 4 = 0$
$\Rightarrow a - 2 = 0,\ b - 3 = 0$ and $c - 4 = 0$
$\Rightarrow a = 2,\ b = 3$ and $c = 4$
$\therefore\ \ \ a + b + c = 2 + 3 + 4 = 9$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.