Question

The roots of the equation $a x^{2}+b x+c=0$, where $a \in R^{+}$, are two consecutive odd positive integers, then

• A. $|b| \leq 4 a$
• B. $|b| \geq 4 a$
• C. $|b|=2 a$
• D. None of these

Answer 1

Answer: (B)

Let the roots be $\alpha$ and $\alpha+2$,
where $\alpha$ is an odd positive integer.
Then, $a a^{2}+b \alpha+c=0\,\,\,$(1)
and $a(\alpha+2)^{2}+b(\alpha+2)+c=0$
$ \Rightarrow a a^{2}+b \alpha+c+(4 a \alpha$ $+4 a+2 b)=0$
$\Rightarrow 2 a(1+\alpha)+b=0 \,\,\,\,$[using (1)]
$\Rightarrow b=-2 a(1+\alpha)$
$\Rightarrow b^{2}=4 a^{2}(1+\alpha)^{2}$
$ \Rightarrow b^{2} \geq 4 a^{2}(1+1)^{2}$
$[\because \alpha \geq 1$ as $\alpha$ is odd positive integer $]$
$\Rightarrow b^{2} \geq 16 a^{2}$
or $|b| \geq 4 a$