Question

If $A$ and $G$ are the arithmetic and geometric means, respectively, of the roots of a quadratic equation then the equation is ___

Answer 1

A.M. of $a$ and $b$ is $A=\frac{a+b}{2}$
$\Rightarrow 2A=a+b$
G.M. of $a$ and $b$ is, $G=\sqrt{ab}$
$\Rightarrow G^2=ab$
As we know that if the sum of roots and product of roots are given then the quadratic equation is
$x^2-\text{(sum of the roots)x+(product of the roots)}=0$
$\Rightarrow x^2-2Ax+G^2=0$
Hence, the required quadratic equation is
$x^2-2Ax+G^2=0.$