Question

Let A, G and H are the A.M., G.M. and H.M. respectively of two unequal positive integers. Then the equation $A{x}^2-|G|x+\frac{H}{4}=0$ has

• A. Both roots as fractions.
• B. At least one root which is a negative fraction.
• C. Exactly one positive root.
• D. At least one root which is an integer.
• E. None of these.

Answer 1

Answer: (B), (C)

The correct options are
B At least one root which is a negative fraction.
C Exactly one positive root.

Let the numbers be
$a,b$
Hence
$A=\frac{a+b}{2}$
$G=\sqrt{ab}$
$H=\frac{2ab}{a+b}$
Hence
$G^2-AH$
$=ab-\frac{2ab}{a+b}.\frac{a+b}{2}$
$=ab-ab$
$=0$
Hence
The roots are
$x=\frac{|G|}{2A}$
Hence
$x=-\frac{G}{2A}$ and $x=\frac{G}{2A}$