# Write the quadratic equation the arithmetic and geometric

Question:

Write the quadratic equation the arithmetic and geometric means of whose roots are A and G respectively.

Solution:

Let the roots of the required quadratic equation be $a$ and $b$.

$\therefore A=\frac{a+b}{2}$ and $G=\sqrt{a b}$

The equation having $a$ and $b$ as its roots is

$x^{2}-x(a+b)+a b=0$

$\Rightarrow x^{2}-2 A x+G^{2}=0 \quad\left[\because A=\frac{a+b}{2}\right.$ and $\left.G=\sqrt{a b}\right]$