Question

The roots of the equation $a x^2 + 3x + 6 = 0$ will be reciprocal to each other if the value of $a$ is:

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

For reciprocal roots, product = 1 $\Rightarrow c/a = 1$, hence $a = c$. Always check both sum and product for consistency.

Answer 1

The Correct Option is A

Let the roots be $r$ and $\frac{1}{r}$ (reciprocal of each other).
Sum of roots = $r + \frac{1}{r} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{3}{a}$.
Product of roots = $r \frac{1}{r} = 1 = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{6}{a}$.
From the product equation: $\frac{6}{a} = 1 \Rightarrow a = 6$. Wait — this gives 6, but check sign conditions: In quadratic, product = $c/a$. For reciprocal roots, $c/a = 1 \Rightarrow c = a$. Here $c = 6$, so $a = 6$. This conflicts with the given choices and sum constraint must also hold for real roots. But since we only need the reciprocal property, $a = 6$ is correct mathematically. If the problem expects $a=c$ rule, answer = 6. But if typographical sign is different, answer may shift. Accepting correct formula: $a = c = 6$.