Question

Let $A,G$ and $H$ are the $AM,GM$ and $HM$ respectively of two unequal positive integers. Then the equation $A{x}^2-|G|x-H=0$ has

• A. both roots as fraction
• B. at least one root which is a negative fraction
• C. exactly one positive root
• D. at least one root which is an integer

Answer 1

Answer: (B), (C)

at least one root which is a negative fraction
exactly one positive root

Let $a$ & $b$ be two positive numbers,
$A=\frac{a+b}{2}$, $G=\sqrt{ab}$, $H=\frac{2ab}{a+b}$
$A{x}^2-|G|x-H=0$
$\Rightarrow \left(\frac{a+b}{2}\right)x^2-\sqrt{ab}x-\frac{2ab}{a+b}=0$
$x=\frac{\sqrt{ab}\pm \sqrt{ab+4ab}}{a+b}=\left(\frac{ab}{a+b}\right)(1\pm \sqrt{5})$
Therefore, one root is positive while other is negative.
Ans: B,C